( 4 + )- ( 4 - )= - Vidyadip

MCQ

$\tan \left(\frac{\pi}{4}+\theta\right)-\tan \left(\frac{\pi}{4}-\theta\right)=$

✓
$2 \tan 2 \theta$
B
$2 \cot 2 \theta$
C
$\tan 2 \theta$
D
$\cot 2 \theta$

✓

Answer

Correct option: A.

$2 \tan 2 \theta$

(A)
$\tan \left(\frac{\pi}{4}+\theta\right)-\tan \left(\frac{\pi}{4}-\theta\right)$
$\begin{aligned}i. \tan \left(45^{\circ}+\theta\right) & =\frac{1+\tan \theta}{1-\tan \theta} \\ & =\frac{\cos \theta+\sin \theta}{\cos \theta-\sin \theta}\end{aligned}$
$\begin{aligned}ii. \tan \left(45^{\circ}-\theta\right) & =\frac{1-\tan \theta}{1+\tan \theta} \\ & =\frac{\cos \theta-\sin \theta}{\cos \theta+\sin \theta}\end{aligned}$
$ =\frac{1+\tan \theta}{1-\tan \theta}-\frac{1-\tan \theta}{1+\tan \theta} $
$=\frac{4 \tan \theta}{1-\tan ^2 \theta}$
$=2\left(\frac{2 \tan \theta}{1-\tan ^2 \theta}\right)=2 \tan 2 \theta$

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