Question

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