Question

If $x=\sin t\mathrm{and}y=\tan t$, then $\frac{dy}{dx}$ is equal to:

• A. ${\cos }^{3}t$
• B. $\frac{1}{{\cos }^{3}t}$
• C. $\frac{1}{{\cos }^{2}t}$
• D. $\frac{1}{{\sin }^{2}t}$

Answer 1

The correct option is B

$\frac{1}{{\cos }^{3}t}$

Explanation for the correct option:

Parametric Differentiation:

$\begin{array}{rcl}x& =& \sin t\\ & \Rightarrow & \frac{dx}{dt}=\cos t\\ y& =& \tan t\\ & \Rightarrow & \frac{dy}{dt}=se{c}^{2}t\\ & =& \frac{1}{{\cos }^{2}t}\end{array}$

Now,

$\begin{array}{rcl}\frac{dy}{dx}& =& \frac{dy}{dt}\times \frac{dt}{dx}\\ & =& \frac{1}{{\cos }^{2}t}\times \frac{1}{\cos t}\\ & =& \frac{1}{{\cos }^{3}t}\end{array}$

Hence, option B is correct.