Question

Consider the following statements
Statement I $\frac{\cos (\pi+x) \cdot \cos (-x)}{\sin (\pi-x) \cdot \cos \left(\frac{\pi}{2}+x\right)}=\cot ^2 x$
Statement II $\cos (\pi+\theta)=-\cos \theta$ and $\cos (-\theta)=\cos \theta$
Also, $\sin (\pi-\theta)=\sin \theta$ and $\sin (-\theta)=-\sin \theta$.
Choose the correct option.

• A. Statement I is true; Statement II is true; Statement II is a correct explanation for Statement I
• B. Statement I is true; Statement II is true; Statement II is not correct explanation for Statement I
• C. Statement I is true; Statement II is false
• D. Statement I is false; Statement II is true

Answer 1

Answer: (A)

$ \frac{\cos (\pi+x) \cos (-x)}{\sin (\pi-x) \cos \left(\frac{\pi}{2}+x\right)}=\frac{(-\cos x)(\cos x)}{(\sin x)(-\sin x)}$
$ \begin{bmatrix} \because \cos (\pi+\theta) & =-\cos \theta \\\cos (-\theta) & =\cos \theta \\\sin (\pi-\theta) & =\sin \theta \\\sin (-\theta) & =-\sin \theta\end{bmatrix} $
$ =\frac{\cos ^2 x}{\sin ^2 x}=\cot ^2 x $
So, both the statements are true and Statement II is the correct explanation of Statement I.