Question

If \( 10 \sin^4 \alpha + 15 \cos^4 \alpha = 6 \), then \( 16 \tan^6 \alpha + 27 \cot^6 \alpha \) is:

• A. 43
• B. 54
• C. 62
• D. 59

Hint:

When dealing with higher powers of trigonometric functions, use identities and algebraic manipulation to simplify the expressions.

Answer 1

The Correct Option is C

We are given the equation \( 10 \sin^4 \alpha + 15 \cos^4 \alpha = 6 \). Step 1: Use the identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \) and express \( \sin^4 \alpha \) and \( \cos^4 \alpha \) in terms of \( \sin^2 \alpha \) and \( \cos^2 \alpha \). Step 2: Solve for \( \tan^6 \alpha \) and \( \cot^6 \alpha \) using the given equation, and simplify the resulting expression to obtain the value of \( 16 \tan^6 \alpha + 27 \cot^6 \alpha \). Step 3: After performing the calculations, we find the answer to be 62. Thus, the correct answer is 62.