If f(x) = 3 \left [ \sin^{4} \left (\frac{3\pi}{2} -x\right )+\sin^{4}(3\pi+x)\right ] - 2 \left [\sin^{6} \left (\frac{\pi}{2}+x\right ) + \sin^{6}( 5\pi-x \right ) ], then for all permissible values of x, f(x) is 

  • Option 1)

    -1

  • Option 2)

    0

  • Option 3)

    1

  • Option 4)

    Not a constant function

 

Answers (1)

 

Allied Angles -

- wherein

The trigonometric ratios for angles in all the four quadrants.

 

 f(x)=3 \left [ \sin ^{4}\left ( \frac{3\pi }{2}-x \right )+\sin ^{4}(3\pi +x) \right ]

-2 \left [ \sin ^{6}\left ( \frac{\pi }{2}+x \right )+\sin ^{6}(5\pi -x) \right ]

\Rightarrow 3(\cos ^{4}x+\sin ^{4}x) - 2 (\cos ^{6}x+\sin ^{6}x)

\Rightarrow 3\left [ (\cos ^{2}x+\sin ^{2}x)^{2} -2\sin ^{2}x \cos^{2}x\right ]

-2\left [ (\sin ^{2}x +\cos ^{2}x)^{3}-3\sin ^{2}x\cos ^{2}x (\sin ^{2}x +\cos ^{2}x) \right ]

\Rightarrow 3\left [ 1-2\sin ^{2}x\cos ^{2}x \right ] - 2 \left [ 1-3\sin ^{2}x\cos ^{2}x \right ]

\Rightarrow 3-6 \sin ^{2}x\cos ^{2}x-2+6\sin ^{2}x\cos ^{2}x

\Rightarrow 3-2 = 1


Option 1)

-1

This is incorrect option

Option 2)

0

This is incorrect option

Option 3)

1

This is correct option

Option 4)

Not a constant function

This is incorrect option

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