First slide

Trigonometric Identities

Question

If $15 \sin^{4} α + 10 \cos^{4} α = 6$ then the value of $8 {cosec}^{6} α - 27 \sec^{6} α$ is

Moderate

A

0
B

1
C

-1
D

5

Solution

$\begin{array}{l} 15 \tan^{4} α + 10 = 6 {(1 + \tan^{2} α)}^{2} \\ \Rightarrow 9 \tan^{4} α - 12 \tan^{2} α + 4 = 0 \\ \Rightarrow \tan^{2} α = 2 / 3 \\ so 8 {cosec}^{6} α - 27 \sec^{6} α \\ = 8 (1 + 3 / 2)^{3} - 27 (1 + 2 / 3)^{3} = 0 \end{array}$

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