Question

I need help in this tricky maths question.

Answer 1

Answer:

$\begin{array}{l}\displaystyle \rm\bigstar \: \: \: \: \[ \frac{1-\tan ^{2}\left(\frac{\pi}{4}+\theta\right)}{1+\tan ^{2}\left(\frac{\pi}{4}+\theta\right)} \] \\ \\ \\ \bigcirc \: \sin \: 2 \theta \\ \red{ \pmb{{ \bigodot \: - \sin \: 2 \theta }}} \\ \bigcirc \: \cos2 \theta \end{array}$

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Answer: B

Step-by-step explanation:

$\\ \rm\bigstar \: \: \: \: \[ \frac{1-\tan ^{2}\left(\frac{\pi}{4}+\theta\right)}{1+\tan ^{2}\left(\frac{\pi}{4}+\theta\right)} \]$

We know that, $\\ \rm \cos 2 \theta=\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}$

$\[ \begin{array}{l} \\ \displaystyle\rm \frac{1-\tan ^{2}\left(\frac{\pi}{4}+\theta\right)}{1+\tan ^{2}\left(\frac{\pi}{4}+\theta\right)} \\\\ \displaystyle\rm =\cos 2\left(\frac{\pi}{4}+\theta\right) \\\\ \displaystyle\rm =\cos \left(\frac{\pi}{2}+2 \theta\right) \\ \\ \boxed{\green{ \displaystyle\rm=-\sin 2 \theta}} \end{array} \]$