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

$10 = \sec^{4} θ + \sec^{2} θ - \tan^{4} θ - \tan^{2} θ$
$= \sec^{2} θ + \tan^{2} θ + 1 = 2 \sec^{2} θ$
$\cos^{2} θ = \frac{1}{5} \Rightarrow \sin^{2} θ = \frac{4}{5}$
$\frac{\sin^{3} θ - \cos^{3} θ}{\sin θ - \cos θ} = \sin^{2} θ + \cos^{2} θ + \sin θ \cos θ,$

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