Question

$\sqrt{(1-{\cos }^2\theta )se{c}^2\theta }=\tan \theta$

• A. True
• B. False

Answer 1

The correct option is A True

Consider,

$$\begin{gathered} L.H.S.=\sqrt{(1-{\cos }^2\theta )se{c}^2\theta } \\ =\sqrt{{\sin }^2\theta \times se{c}^2\theta }.......({\sin }^2\theta +{\cos }^2\theta =1) \\ =\sqrt{{\sin }^2\theta \times \frac{1}{{\cos }^2\theta }} \\ =\sqrt{\frac{{\sin }^2\theta }{{\cos }^2\theta }}......(\frac{\sin \theta }{\cos \theta }=\tan \theta ) \\ =\sqrt{{\tan }^2\theta } \\ =\tan \theta \end{gathered}$$

$R.H.S.=\tan \theta$

Therefore, $\sqrt{(1-{\cos }^2\theta )se{c}^2\theta }=\tan \theta$ is a true statement