Question

Consider the following statements
Statement I $\cot 4 x(\sin 5 x+\sin 3 x)$
$=\cot x(\sin 5 x-\sin 3 x)$
Statement II $\sin 5 x+\sin 3 x=2 \sin 4 x \cdot \cos x$
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 a 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: (B)

I. $\cot 4 x(\sin 5 x+\sin 3 x)$
$=\cot 4 x \cdot 2 \sin \frac{5 x+3 x}{2} \cos \frac{5 x-3 x}{2} $
$ \left(\because \sin C+\sin D=2 \sin \frac{C+D}{2} \cos \frac{C-D}{2}\right)$
$ =\frac{\cos 4 x}{\sin 4 x} \times 2 \sin 4 x \cos x$
$=2 \cos 4 x \cos x$
Now, $\cot x(\sin 5 x-\sin 3 x)$
$=\cot x \cdot 2 \cos \frac{5 x+3 x}{2} \sin \frac{5 x-3 x}{2}$
$\left(\because \sin C-\sin D=2 \cos \frac{C+D}{2} \sin \frac{C-D}{2}\right)$
$=\frac{\cos x}{\sin x} \times 2 \cos 4 x \sin x=2 \cos 4 x \cos x$
$\therefore LHS = RHS$
Hence, Statement I is true.
Also, Statement II is also true but not a correct explanation of Statement I.