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


Mathematics

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