First slide

Trigonometric Identities

Question

If $\cot θ + \tan θ = x and \sec θ - \cos θ = y,$ then

Moderate

A

$xsin θ \cdot \cos θ = 1$
B

$\sin^{2} θ = ycos θ$
C

${(x^{2} y)}^{1 / 3} + {({xy}^{2})}^{1 / 3} = 1$
D

${(x^{2} y)}^{2 / 3} - {({xy}^{2})}^{2 / 3} = 1$

Solution

$\cot θ + \tan θ = x$
$\begin{array}{l} or \frac{\cos θ}{\sin θ} + \frac{\sin θ}{\cos θ} = x \\ or 1 = xsin θcos θ \end{array}$
$Now \sec θ - \cos θ = y$
$\begin{aligned} \frac{1}{\cos θ} - \cos θ = y \\ or \sin^{2} θ = ycos θ \end{aligned}$
$\begin{array}{l} now x^{2} y = \frac{1}{\sin^{2} {θcos}^{2} θ} \cdot \frac{\sin^{2} θ}{\cos θ} = \frac{1}{\cos^{3} θ} \\ and {xy}^{2} = \frac{\sin^{3} θ}{\cos^{3} θ} \\ ∴ {(x^{2} y)}^{2 / 3} - {({xy}^{2})}^{2 / 3} = \frac{1}{\cos^{2} θ} - \frac{\sin^{2} θ}{\cos^{2} θ} = 1 \end{array}$

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