Question

Assertion :$\int \tan 3x\tan 2x\tan xdx=-\frac{\log \left|\cos 3x\right|}{3}+\frac{\log \left|\cos 2x\right|}{2}+\log \left|\cos x\right|+C$ Reason: $\tan 3x\tan 2x\tan x=\tan 3x-\tan 2x-\tan x$

• 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

$\because$ $3x=2x+x$
$\therefore \tan 3x=\frac{\tan 2x+\tan x}{1-\tan 2x\tan x}$
$\Rightarrow \tan 3x\tan 2x\tan x=\tan 3x-\tan 2x-\tan x$
$\therefore$ Reason (R) is true
$\Rightarrow \int \tan 3x\tan 2x\tan xdx=\int (\tan 3x-\tan 2x-\tan x)dx$
$=\frac{-\log \left|\cos 3x\right|}{3}+\frac{\log \left|\cos 2x\right|}{2}+\log \left|\cos x\right|+C$
Assertion (A) is correct & Reason (R) is correct and Reason is the correct explanation of Assertion.