Question

Assertion :$\int \tan 5x.\tan 3x.\tan 2xdx=\frac{\ln \left|\sec 5x\right|}{5}-\frac{\ln \left|\sec 3x\right|}{3}-\frac{\ln \left|\sec 2x\right|}{2}+C$ Reason: $\tan 5x-\tan 3x-\tan 2x=\tan 5x.\tan 3x.\tan 2x$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$\tan 5x=\tan (2x+3x)$
$\tan 5x=\frac{\tan 2x+\tan 3x}{1-\tan 2x\tan 3x}$
$\Rightarrow \tan 5x-\tan 2x\tan 3x\tan 5x=\tan 2x+\tan 3x$
$\Rightarrow \tan 5x-\tan 2x-\tan 3x=\tan 2x\tan 3x\tan 5x$
$\int \tan 2x\tan 3x\tan 5xdx=\int \tan 5xdx-\int \tan 2xdx-\int \tan 3xdx$
$=\frac{ln|\sec 5x|}{5}-\frac{ln|\sec 2x|}{2}\frac{ln|\sec 3x|}{3}$
Hence, assertion and reason both are correct. Reason is the correct explanation of assertion