First slide

Trigonometric Identities

Question

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

Moderate

A

0
B

1
C

2
D

3

Solution

$\begin{aligned} \sec x = 1 - \sec^{2} x \\ \Rightarrow \sec x = - \tan^{2} x \end{aligned}$
Squaring both side, we get
$\begin{array}{l} \sec^{2} x = \tan^{4} x \\ \Rightarrow 1 + \tan^{2} x = \tan^{4} x \end{array}$
Squaring both side, we get
$\begin{array}{l} \tan^{8} x = \tan^{4} x + 2 \tan^{2} x + 1 \\ \Rightarrow \tan^{8} x - \tan^{4} x - 2 \tan^{2} x = 1 \\ \Rightarrow \tan^{8} x - \tan^{4} x - 2 \tan^{2} x + 1 = 2 \end{array}$

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