Cos4x = 1-8sin square x cos2x
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Answer:
Given cos4x=1−8sin
2
xcos
2
x
Taking LHS
cos4x
We know
cos2x=2cos
2
x−1
Replacing x by 2x
cos2(2x)=2cos
2
(2x)−1
cos4x=2cos
2
2x−1
=2(cos2x)
2
−1
Using cos2x=2cos
2
x−1
=2(2cos
2
x−1)
2
−1
Using (a−b)
2
=a
2
+b
2
−2ab
=2[(2cosx)
2
+(1)
2
−2(2cos
2
x)×1]−1
=2(4cos
4
x+1−4cos
2
x)−1
=2×4cos
4
x+2−2×4cos
2
x−1
=8cos
4
x+2−8cos
2
x−1
=8cos
4
x−8cos
2
x+2−1
=8cos
2
x(cos
2
x−1)+1
=8cos
2
x[−(1−cos
2
x)]+1
=−8cos
2
x[(1−cos
2
x)]+1
=−8cos
2
xsin
2
x+1 [∵sin
2
x=1−cos
2
x]
=1−8cos
2
xsin
2
x
=RHS
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