First slide

Multiple and sub- multiple Angles

Question

If $(1 + \sqrt{1 + x}) \tan â¡ Î± = (1 â \sqrt{1 â x})$ then $x =$

Moderate

A

$\sin a$
B

$\sin 2 a$
C

sin 4a
D

$\cos 4 a$

Solution

$\begin{array}{l} (1 + \sqrt{1 + x}) \tan â¡ Î± = 1 â \sqrt{1 â x} \\ â \frac{1 + \sqrt{1 + x}}{1 â \sqrt{1 â x}} = \frac{\cos â¡ Î±}{\sin â¡ Î±} = \frac{2 \cos â¡ Î± (\cos â¡ Î± + \sin â¡ Î±)}{2 \sin â¡ Î± (\cos â¡ Î± + \sin â¡ Î±)} \\ = \frac{2 \cos^{2} â¡ Î± + 2 \sin â¡ Î± \cos â¡ Î±}{2 \sin â¡ Î± \cos â¡ Î± + 2 \sin^{2} â¡ Î±} \\ = \frac{1 + \cos â¡ 2 Î± + \sin â¡ 2 Î±}{\sin â¡ 2 Î± + 1 â \cos â¡ 2 Î±} \\ = \frac{1 + \sqrt{(\cos â¡ 2 Î± + \sin â¡ 2 Î±)^{2}}}{1 â \sqrt{(\cos â¡ 2 Î± â \sin â¡ 2 Î±)^{2}}} \\ = \frac{1 + \sqrt{(1 + \sin â¡ 4 Î±)}}{1 â \sqrt{1 â \sin â¡ 4 Î±}} \end{array}$

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