First slide

Multiple and sub- multiple Angles

Question

$If \sin x + \cos x + \tan x + cosec x + \sec x + \cot x = 7, then \sin 2 x =$

Moderate

A

$22 + 8 \sqrt{7}$
B

$22 - 8 \sqrt{7}$
C

11
D

22

Solution

$(\sin x + \cos x) + \frac{1}{\sin x \cos x} + \frac{\sin x + \cos x}{\sin x \cos x} = 7$
$\Rightarrow (\sin x + \cos x) (1 + \frac{1}{\sin x \cos x}) = 7 - \frac{1}{\sin x \cos x}$
Squaring on both sides we get
$\begin{array}{l} \Rightarrow \sin 2 x (\sin^{2} 2 x - 44 \sin 2 x + 36) = 0 \\ \sin 2 x \neq 0 \\ \Rightarrow \sin 2 x = 22 - 8 \sqrt{7} \end{array}$

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