Q. 16

# Mark the Correct alternative in the following:The value of 2 cos x – cos 3x – cos 5x – 16 cos3 x sin2 x isA.2B. 1C. 0D. -1

Given expression is 2 cos x – cos 3x – cos 5x – 16 cos3 x sin2 x

Consider the expression

2 cos x – cos 3x – cos 5x – 16 cos3 x sin2

= 2 cos x – (cos 5x + cos 3x) – 16 cos3 x sin2 x

[using the sum of angles ]

= 2 cos x – [2 cos 4x cos x] - 16cos3 x sin2 x

= 2 cos x (1 – cos 4x ) - 16cos3 x sin2 x

[ using the property cos 2θ = 1- 2 sin2 θ ]

= 2 cos x [ 1 – (1 – 2 sin2 2x)] -16cos3 x sin2 x

= 2 cos x [2 sin2 2x] -16cos3 x sin2 x

= 4cos x [2sin x cos x]2 -16cos3 x sin2 x

[ using sin 2θ = 2 sin θ cos θ ]

= 4 × 4 (cos x sin2 x cos2 x) -16cos3 x sin2 x

= 16cos3 x sin2 x -16cos3 x sin2 x

= 0

Hence cos x – cos 3x – cos 5x – 16 cos3 x sin2 x = 0

The answer is option C.

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