Solve the 3 Question and please prove it true or false.
Q.3. $\sqrt{\left(1-{\cos }^2\theta \right)se{c}^2\theta }=\tan \theta$

Please find below the solution to the asked query:

We have : $\sqrt{\left(1 - {\mathrm{Cos}}^2 \mathrm{\theta }\right){\mathrm{Sec}}^2 \mathrm{\theta }} = \tan \mathrm{\theta }$ , TakeL.H.S. :

$$\begin{gathered} \sqrt{\left(1 - {\mathrm{Cos}}^2 \mathrm{\theta }\right){\mathrm{Sec}}^2 \mathrm{\theta }} \\ \mathrm{We} \mathrm{know} : \mathrm{Cos} \mathrm{\theta } = \frac{1}{\mathrm{Sec} \mathrm{\theta }} , \mathrm{So} \\ \Rightarrow \sqrt{\left(1 - \frac{1}{{\mathrm{Sec}}^2 \mathrm{\theta }}\right){\mathrm{Sec}}^2 \mathrm{\theta }} \\ \Rightarrow \sqrt{\left(\frac{{\mathrm{Sec}}^2 \mathrm{\theta } - 1}{{\mathrm{Sec}}^2 \mathrm{\theta }}\right){\mathrm{Sec}}^2 \mathrm{\theta }} \\ \Rightarrow \sqrt{{\mathrm{Sec}}^2 \mathrm{\theta } - 1} \\ \mathrm{We} \mathrm{know} : 1 + {\tan }^2 \mathrm{\theta } = {\mathrm{Sec}}^2 \mathrm{\theta } , \mathrm{So} \\ \Rightarrow \sqrt{{\tan }^2 \mathrm{\theta } } \\ \mathbf{\Rightarrow }\mathbf{tan}\mathbf{ }\mathbf{\theta } \end{gathered}$$
So, L.H.S. = R.H.S.

Therefore,

Given equation is true .

Hope this information will clear your doubts about topic.