First slide

Trigonometric Identities

Question

If $\sec x + \sec^{2} x = 1$ then the value of $\tan^{8} x - \tan^{4} x - 2 \tan^{2} x + 1$ is equal to

Moderate

A

0
B

1
C

2
D

3

Solution

$\begin{array}{ll} We have, & \sec x + \sec^{2} x = 1 \\ \sec x = - (\sec^{2} x - 1) = - \tan^{2} x \\ \Rightarrow & \sec^{2} x = \tan^{4} x \Rightarrow 1 + \tan^{2} x = \tan^{4} x \\ \Rightarrow & {(1 + \tan^{2} x)}^{2} = \tan^{8} x \\ \Rightarrow & 1 + 2 \tan^{2} x + \tan^{4} x = \tan^{8} x \\ \Rightarrow & \tan^{8} x - \tan^{4} x - 2 \tan^{2} x = 1 \\ \Rightarrow & \tan^{8} x - \tan^{4} x - 2 \tan^{2} x + 1 = 2 \end{array}$

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