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Question: \[\tan\left( \frac{\pi}{4} + \theta \right) - \tan\left( \frac{\pi}{4} - \theta \right) =\]...

tan⁡(π4+θ)−tan⁡(π4−θ)=\tan\left( \frac{\pi}{4} + \theta \right) - \tan\left( \frac{\pi}{4} - \theta \right) =

A

2tan⁡2θ2\tan 2\theta

B

2cot⁡2θ2\cot 2\theta

C

tan⁡2θ\tan 2\theta

D

cot⁡2θ\cot 2\theta

Answer

2tan⁡2θ2\tan 2\theta

Explanation

Solution

tan⁡(π4+θ)−tan⁡(π4−θ)=1+tan⁡θ1−tan⁡θ−1−tan⁡θ1+tan⁡θ\tan\left( \frac{\pi}{4} + \theta \right) - \tan\left( \frac{\pi}{4} - \theta \right) = \frac{1 + \tan\theta}{1 - \tan\theta} - \frac{1 - \tan\theta}{1 + \tan\theta}

=4tan⁡θ1−tan⁡2θ=2(2tan⁡θ1−tan⁡2θ)=2tan⁡2θ= \frac{4\tan\theta}{1 - \tan^{2}\theta} = 2\left( \frac{2\tan\theta}{1 - \tan^{2}\theta} \right) = 2\tan 2\theta.