Question

Derivative of ${\tan }^3\theta$ with respect to ${\sec }^3\theta$ at $\theta =\frac{\pi }{3}$ is _____

• A. $\frac{3}{2}$
• B. $\frac{\sqrt{3}}{2}$
• C. $\frac{1}{2}$
• D. $-\frac{\sqrt{3}}{2}$

Answer 1

The correct option is B $\frac{\sqrt{3}}{2}$

Derivative of $ta{n}^3\theta$ wrt $\theta$ is $3ta{n}^2\theta se{c}^2\theta$

Derivative of $se{c}^3\theta$ wrt $\theta$ is $3se{c}^3\theta tan\theta$

Therefore the derivative of $ta{n}^3\theta$ with respect to $se{c}^3\theta$ will be $\frac{3ta{n}^2\theta se{c}^2\theta }{3tan\theta se{c}^3\theta }=\frac{tan\theta }{sec\theta }=sin\theta$

The value of $sin\theta$ at $\theta =\frac{\pi }{3}$ will be $\frac{\sqrt{3}}{2}$

Therefore the correct option is $B$