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Question

Total number of solutions of equation sinx.tan4x=cosx belonging to (π,2π) are

A
4
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B
7
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C
8
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D
12
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Solution

The correct option is D 12
We have sinxtan4x=cosx

sinxcosxtan4x=1

tanxtan4x=1

tanxtan2(2x)=1

2tan2x1tan22x=1tanx

2(2tanx1tan2x)1(2tanx1tan2x)2=1tanx

2(2tanx1tan2x)(1tan2x)24tan2x(1tan2x)2=1tanx

4tanx(1tan2x)12tan2x+tan4x4tan2x=1tanx

4tanx4tan3x16tan2x+tan4x=1tanx

4tan2x4tan4x=16tan2x+tan4x

5tan4x10tan2x+1=0

So, tan2x=10±1002010=10±4510

tan2x=5255 or tan2x=5+255

tanx=±(5255)1/2 or tanx=±(5+255)1/2

As x(π,2π), tanx has two positive and two negative values.

Also tanx>0x(π,π2)(0,π2)(π,3π2)

and tanx<0x(π2,0)(π2,π)(3π2,2π)

So the number of values of x which satisfy the given equation is 12.

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