Answer :
The answer to a power reduction to simply 8cos^4x is 2(cos2a + 1)^2
The simplified value is 8*cos^4(x) = 3+ 4 cos2x + cos 4x.
What is power reduction identity?
The trigonometric power reduction identities allow us to rewrite expressions involving trigonometric terms with trigonometric terms of smaller powers. This becomes important in several applications such as integrating powers of trigonometric expressions in calculus.
Given:
Using the power reduction identity, we have that:
cos² x= 1/2( 1+ cos 2x)
cos^4(x) = (cos² x)²= (1/2( 1+ cos 2x))²
cos^4(x) = 1/4( 1+ cos² 2x+ 2 cos (2x))
Again, cos² 2x= 1/2( 1+ cos 4x)
cos^4(x) = 1/4( 1+ 1/2( 1+ cos 4x)+ 2 cos (2x))
cos^4(x) = 1/4( 1+ 1/2 +1/2 cos 4x+ 2 cos (2x))
cos^4(x) = 1/4( 3/2 + 2 cos2x + 1/2 cos 4x)
8*cos^4(x) = 8*1/4( 3/2 + 2 cos2x + 1/2 cos 4x)
8*cos^4(x) = 2( 3/2 + 2 cos2x + 1/2 cos 4x)
8*cos^4(x) = 3+ 4 cos2x + cos 4x
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