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Question

Answers

A ) $\pi $

B ) $n\pi $

C ) $2n\pi $

D ) $\left( {2n + 1} \right)\pi $

Answer
Verified

Let us assume that the angle is $\theta $.

Thus, from the given statement in the question we can write,

$\dfrac{d}{{d\theta }}\left( {\sin \theta } \right) = \dfrac{d}{{d\theta }}\left( {\tan \theta } \right)$

Using the formula of differentiation of standard trigonometric terms, we get

$\cos \theta = {\sec ^2}\theta $

We know that $\sec \theta = \dfrac{1}{{\cos \theta }}$ , thus, the above equation can be written as

$\begin{array}{l}

\cos \theta = \dfrac{1}{{{{\cos }^2}\theta }}\\

\Rightarrow {\cos ^3}\theta = 1

\end{array}$

Taking cube roots on both sides we get

$\cos \theta = 1$

We know that for a particular value in $\left[ {0,2\pi } \right]$ , $\cos 0 = 1$

Since no particular value is required in this question, we can write the right hand side of the above equation as

$\begin{array}{l}

1 = \cos \left( {2n\pi + 0} \right)\;,n \in I\\

\Rightarrow 1 = \cos \left( {2n\pi } \right)\;,n \in I

\end{array}$

We know from general solution of trigonometric equations we get

$\begin{array}{l}

\cos \theta = \cos \left( {2n\pi } \right)\\

\Rightarrow \theta = 2n\pi \;,\;n \in I

\end{array}$

Thus, the third option is the correct option.

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