Question

Assertion :$\tan \alpha +2\tan 2\alpha +4\tan 4\alpha +8\tan 8\alpha +16\cot 16\alpha =\cot \alpha$ Reason: $\cot \alpha -\tan \alpha =2\cot 2\alpha$

• A. Statement-1 is false, statement-2 is true
• B. Statement-1 is true, statement-2 is true,statement-2 is a correct explanation for statement-1
• C. Statement-1 is true, statement-2 is true,statement-2 is not a correct explanation for statement-1
• D. Statement-1 is true, statement-2 is false

Answer 1

The correct option is B Statement-1 is true, statement-2 is true,statement-2 is a correct explanation for statement-1

$\cot \theta -\tan \theta =\frac{1-{\tan }^2\theta }{\tan \theta }=2\left(\frac{1-{\tan }^2\theta }{2\tan \theta }\right)=2\cot 2\theta$
$\therefore \tan \alpha +2\tan 2\alpha +4\tan 4\alpha +8\tan 8\alpha +16\cot 16\alpha =-(\cot \alpha -\tan \alpha -2\tan 2\alpha -4\tan 4\alpha -8\tan 8\alpha )+16\cot 16\alpha +\cot \alpha =-(2\cot 2\alpha -2\tan 2\alpha -4\tan 4\alpha -8\tan 8\alpha )+16\cot 16\alpha +\cot \alpha =-(4\cot 4\alpha -4\tan 4\alpha -8\tan 8\alpha )+16\cot 16\alpha +\cot \alpha =-(8\cot 8\alpha -8\tan 8\alpha )+16\cot 16\alpha +\cot \alpha =-16\cot 16\alpha +16\cot 16\alpha +\cot \alpha =\cot \alpha$