Prove Cosec 10 Degrees - √3 Sec 10 Degrees Equals 4 A Trigonometric Proof

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Introduction

In trigonometry, proving trigonometric identities often involves manipulating trigonometric functions to show that two expressions are equal. This requires a deep understanding of trigonometric relationships and the ability to apply various trigonometric formulas. In this article, we aim to demonstrate that the expression cosec 10° - √3 sec 10° is indeed equal to 4. This demonstration will not only showcase the fundamental trigonometric identities but also highlight how these identities can be strategically used to simplify complex trigonometric expressions.

The problem at hand, proving that cosec 10° - √3 sec 10° = 4, is an intriguing challenge that requires a solid grasp of trigonometric identities and algebraic manipulation. The key to solving this problem lies in converting the given expression into a more manageable form using fundamental trigonometric identities. Specifically, we will convert cosecant and secant functions into their sine and cosine counterparts, respectively. This conversion allows us to combine the terms into a single fraction, which can then be further simplified. This initial step is crucial as it sets the stage for subsequent manipulations involving angle sum and difference identities. The use of these identities will enable us to break down the trigonometric functions of complex angles into simpler terms, making the equation more amenable to simplification.

This article delves into a step-by-step solution, providing a clear and concise explanation of the methods used. By understanding these methods, readers can enhance their problem-solving skills and deepen their appreciation for the elegance of trigonometric manipulations. Our approach will not only focus on the mathematical steps but also emphasize the logical reasoning behind each step, thereby making the solution accessible and comprehensible. The final solution will not only validate the given trigonometric identity but also offer insights into the broader application of trigonometric principles in mathematical problem-solving. Through a combination of algebraic manipulation and trigonometric identities, we will transform the initial complex expression into a simple numerical value, thus completing the proof.

Step-by-Step Proof

To prove the trigonometric identity cosec 10° - √3 sec 10° = 4, we will start by expressing cosecant and secant in terms of sine and cosine, respectively. This initial step is pivotal in transforming the expression into a form that is easier to manipulate. The reciprocal identities, which state that cosec θ = 1/sin θ and sec θ = 1/cos θ, are the cornerstone of this transformation. By applying these identities, we effectively shift our focus from cosecant and secant functions to sine and cosine functions, which are more commonly used in trigonometric manipulations. This conversion not only simplifies the expression but also allows us to leverage various sine and cosine identities to further our proof.

1. Express Cosecant and Secant in Terms of Sine and Cosine

We begin by rewriting the given expression using the reciprocal identities:

cosec 10° - √3 sec 10° = (1 / sin 10°) - √3 (1 / cos 10°)

This transformation is a standard technique in trigonometry, allowing us to work with more fundamental trigonometric functions. By expressing cosecant and secant in terms of sine and cosine, we pave the way for combining the terms into a single fraction. This step is critical in simplifying the expression and making it more manageable for subsequent trigonometric manipulations. The use of reciprocal identities is a common strategy in proving trigonometric identities, as it often simplifies the equation and makes it easier to apply other identities.

2. Combine the Terms into a Single Fraction

To combine the terms, we find a common denominator, which in this case is sin 10° cos 10°. This process involves multiplying the first term by cos 10° / cos 10° and the second term by sin 10° / sin 10°. This step is crucial in bringing the terms under a single fraction, thereby simplifying the expression and making it easier to manipulate. The combined fraction allows us to apply trigonometric identities that involve both sine and cosine, which are essential for further simplification.

= (cos 10° - √3 sin 10°) / (sin 10° cos 10°)

This resulting fraction is now in a form where we can apply more advanced trigonometric techniques. The next step involves strategically multiplying the numerator and denominator by a constant to create terms that can be simplified using angle sum or difference identities. This approach is a common strategy in proving trigonometric identities, as it allows us to transform complex expressions into simpler, more manageable forms.

3. Multiply Numerator and Denominator by 2

Next, we multiply both the numerator and the denominator by 2. This step might seem arbitrary at first, but it is a crucial maneuver to bring the expression closer to a form where we can apply trigonometric identities effectively. Multiplying by 2 does not change the value of the expression, but it sets the stage for further simplification by allowing us to create terms that match the structure of angle sum or difference formulas. This technique is often employed in trigonometric proofs to manipulate expressions into a more recognizable form.

= 2 (cos 10° - √3 sin 10°) / (2 sin 10° cos 10°)

The denominator now contains the term 2 sin 10° cos 10°, which can be simplified using the double-angle identity for sine. This simplification is a key step in reducing the complexity of the expression. Meanwhile, the numerator is primed for further manipulation, which will involve rewriting the terms in a way that allows us to apply angle sum or difference identities. This strategic multiplication by 2 is a common technique in trigonometric proofs, showcasing the importance of algebraic manipulation in conjunction with trigonometric identities.

4. Apply the Double-Angle Identity for Sine

In this step, we apply the double-angle identity for sine, which states that 2 sin θ cos θ = sin 2θ. This identity is a fundamental tool in simplifying trigonometric expressions, especially when dealing with products of sine and cosine functions. Applying this identity to the denominator transforms 2 sin 10° cos 10° into sin 20°, which significantly simplifies the expression. This transformation is a crucial step in our proof, as it reduces the complexity of the denominator and brings us closer to the final solution.

= 2 (cos 10° - √3 sin 10°) / sin 20°

The denominator is now in a much simpler form, which allows us to focus on the numerator. The next step involves manipulating the numerator to create terms that align with the sine or cosine of a sum or difference of angles. This often involves multiplying by a constant and rewriting terms in a way that reveals the underlying trigonometric structure. The strategic use of trigonometric identities, such as the double-angle identity, is essential in simplifying complex expressions and proving trigonometric identities.

5. Multiply Numerator and Denominator by 2 Again

We multiply the numerator and the denominator by 2 once more. This step is crucial for shaping the numerator into a form that allows us to use the sine subtraction formula. Multiplying by 2 helps us create coefficients that align with the standard form of the sine subtraction formula, which is sin(a - b) = sin a cos b - cos a sin b. This maneuver is a common technique in trigonometric manipulations, where strategic multiplication and division are used to transform expressions into more recognizable forms.

= 4 (cos 10° - √3 sin 10°) / (2 sin 20°)

The numerator is now 4 (cos 10° - √3 sin 10°), and the denominator is 2 sin 20°. This form is closer to the application of the sine subtraction formula. The next step involves rewriting the terms in the numerator to match the structure of the sine subtraction formula, which will allow us to simplify the expression further. This strategic manipulation is a key element in proving trigonometric identities, showcasing the interplay between algebraic techniques and trigonometric principles.

6. Rewrite the Numerator to Match the Sine Subtraction Formula

Now, we rewrite the numerator to fit the form sin(A - B) = sin A cos B - cos A sin B. To achieve this, we recognize that √3 can be expressed as tan 60°, which is sin 60° / cos 60°. Thus, we aim to express the numerator in terms of sin 60° and cos 60°. This conversion is a pivotal step, as it directly leads to the application of the sine subtraction formula, simplifying the expression significantly.

We can rewrite the numerator as:

4 (cos 10° - √3 sin 10°) = 4 (2) [(1/2) cos 10° - (√3/2) sin 10°]

Here, we factor out a 2 to create the coefficients 1/2 and √3/2, which are the cosine and sine of 60°, respectively. This manipulation is a strategic move to align the terms with the sine subtraction formula. By recognizing the connection between the coefficients and trigonometric values, we can transform the expression into a form that is readily simplified.

7. Apply Sine Subtraction Formula

Using the values sin 60° = √3/2 and cos 60° = 1/2, we can rewrite the expression inside the brackets as:

= 8 [(1/2) cos 10° - (√3/2) sin 10°] = 8 [cos 60° cos 10° - sin 60° sin 10°]

This form perfectly matches the cosine addition formula cos(A + B) = cos A cos B - sin A sin B. However, a more direct simplification can be achieved by recognizing that the expression is the sine subtraction formula in reverse. This recognition is a key insight that allows us to apply the formula directly and simplify the expression efficiently.

Now, applying the sine subtraction formula, we get:

= 8 sin(60° - 10°) = 8 sin 50°

This simplification is a significant step forward, as it reduces the complexity of the numerator considerably. The use of the sine subtraction formula is a powerful tool in trigonometric manipulations, and its application here demonstrates the elegance of trigonometric identities in simplifying complex expressions.

8. Simplify the Expression

Substituting the simplified numerator back into the expression, we have:

[8 sin 50°] / [2 sin 20°]

Now, we need to express sin 50° in terms of sin 20° to further simplify the expression. We can use the identity sin(90° - θ) = cos θ to rewrite sin 50° as cos 40°. This transformation is a strategic move, as it allows us to leverage the relationship between sine and cosine of complementary angles. By expressing sin 50° as cos 40°, we can then use the double-angle formula for cosine to relate it to sin 20°.

= 4 sin 50° / sin 20°

Next, we use the identity sin(90° - θ) = cos θ:

= 4 cos 40° / sin 20°

9. Use Double Angle Formula

We can express cos 40° using the double-angle formula for cosine, which is cos 2θ = 1 - 2 sin² θ. Applying this formula with θ = 20°, we get:

cos 40° = 1 - 2 sin² 20°

Substituting this back into the expression:

= 4 (1 - 2 sin² 20°) / sin 20°

This form is not directly simplifying to our desired result. Instead, let’s go back to 4 cos 40° / sin 20° and use the identity sin 3θ = 3 sin θ - 4 sin³ θ.

10. Alternative Simplification Path

Recall that sin 30° = 1/2. Using the triple-angle identity for sine:

sin 3θ = 3 sin θ - 4 sin³ θ

Let θ = 10°:

sin 30° = 3 sin 10° - 4 sin³ 10°

1/2 = sin 30° = 3 sin 10° - 4 sin³ 10°

This approach doesn't seem to directly simplify our expression either. Let’s reconsider our original approach of simplifying 4 cos 40° / sin 20°.

11. Revisit and Simplify

We have 4 cos 40° / sin 20°. We can rewrite cos 40° as sin(90° - 40°) = sin 50°. So, we have:

4 sin 50° / sin 20°

Using the sine addition formula, sin(A + B) = sin A cos B + cos A sin B, we can express sin 50° as sin(30° + 20°). This approach seems promising as it introduces the known value of sin 30°.

= 4 [sin(30° + 20°)] / sin 20°

Applying the sine addition formula:

= 4 [sin 30° cos 20° + cos 30° sin 20°] / sin 20°

Substituting sin 30° = 1/2 and cos 30° = √3/2:

= 4 [(1/2) cos 20° + (√3/2) sin 20°] / sin 20°

= 2 [cos 20° + √3 sin 20°] / sin 20°

This form does not seem to simplify directly to 4. We need to backtrack and check for any errors in our steps.

12. Correct Simplification

Going back to 4 cos 40° / sin 20°, we made an error in applying the sine addition formula earlier. Let's reconsider this step.

We want to show 4 cos 40° / sin 20° = 4. This is equivalent to showing cos 40° = sin 20°. However, this is incorrect since cos 40° = sin(90° - 40°) = sin 50°, and sin 50° ≠ sin 20°.

We need to go back to our earlier steps to identify where the error occurred.

13. Identify the Error

Let's review our steps:

  1. cosec 10° - √3 sec 10° = (1 / sin 10°) - √3 (1 / cos 10°)
  2. = (cos 10° - √3 sin 10°) / (sin 10° cos 10°)
  3. = 2 (cos 10° - √3 sin 10°) / (2 sin 10° cos 10°)
  4. = 2 (cos 10° - √3 sin 10°) / sin 20°
  5. = 4 (cos 10° - √3 sin 10°) / (2 sin 20°)
  6. We want to rewrite cos 10° - √3 sin 10° as A sin(B - 10°). Expanding this, we get A (sin B cos 10° - cos B sin 10°). We want A sin B = 1 and A cos B = √3. Dividing, we get tan B = 1/√3, so B = 30°. Then A sin 30° = 1, so A (1/2) = 1, which gives A = 2. So, cos 10° - √3 sin 10° = 2 sin(30° - 10°) = 2 sin 20°.
  7. Substituting this back, we get 4 (2 sin 20°) / (2 sin 20°) = 4.

Thus, the error was in our simplification approach after step 5. The correct simplification is as follows:

= 4 (cos 10° - √3 sin 10°) / (2 sin 20°)

Rewrite the numerator using the sine subtraction formula. We need to find A and B such that:

cos 10° - √3 sin 10° = A sin(B - 10°) = A (sin B cos 10° - cos B sin 10°)

Comparing coefficients, we have:

A sin B = 1

A cos B = √3

Dividing the two equations, we get tan B = 1/√3, which means B = 30°.

Substituting B = 30° into the first equation, we get A sin 30° = 1, so A (1/2) = 1, which means A = 2.

Therefore, cos 10° - √3 sin 10° = 2 sin(30° - 10°) = 2 sin 20°

Substituting this back into the expression:

= 4 (2 sin 20°) / (2 sin 20°) = 4

Conclusion

By carefully applying trigonometric identities and algebraic manipulations, we have successfully proven that cosec 10° - √3 sec 10° = 4. The key to solving this problem was recognizing the structure of trigonometric identities and using them strategically to simplify the expression. This exercise not only validates the given identity but also reinforces the importance of trigonometric principles in mathematical problem-solving.

Final Answer

The final answer is 4.