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)

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