Question

# If $$\sin x+\cos x+\tan x+\cot x+\sec x+\mathrm{cosec} x=7$$ and $$\sin 2x=a-b\sqrt 7$$, then-

A
a=8
B
b=22
C
a=22
D
b=8

Solution

## The correct options are C $$a=22$$ D $$b=8$$$$(\sin x+ \cos x)+(\dfrac {1}{\sin x \cos x})+(\dfrac {\sin x+ \cos x}{\sin x \cos x})=7$$ $$\Rightarrow (\sin x+ \cos x)(1+\dfrac {1}{\sin x \cos x})=7-\dfrac {2}{\sin 2x}$$ Squaring both sides $$\Rightarrow \sin ^2 2x-44 \sin ^22x+36 \sin 2x=0$$ $$\Rightarrow \sin ^2 2x-44 \sin 2x+36=0 (\sin 2x\neq 0)$$ $$\Rightarrow \sin 2x=22-8\sqrt 7$$. Option C and DMathematics

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