First slide

Trigonometric Identities

Question

If $\sec^{4} θ + \sec^{2} θ = 10 + \tan^{4} θ + \tan^{2} θ$ then $\sin^{2} θ =$

Moderate

A

$\frac{2}{3}$
B

$\frac{3}{4}$
C

$\frac{4}{5}$
D

$\frac{5}{6}$

Solution

$\begin{array}{l} \sec^{4} θ + \sec^{2} θ = 10 + \tan^{4} θ + \tan^{2} θ \\ \Rightarrow \sec^{4} θ - \tan^{4} θ + \sec^{2} θ - \tan^{2} θ = 10 \\ \Rightarrow \sec^{2} θ + \tan^{2} θ + 1 = 10 \\ \Rightarrow 2 \sec^{2} θ = 10 \\ \Rightarrow \cos^{2} θ = \frac{1}{5} \\ \Rightarrow \sin^{2} θ = \frac{4}{5} \end{array}$

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