Question #73def

1 Answer
yumean1119
Dec 2, 2017

Apply the double-angle formulae to prove this.

Explanation:

The double-angle formula for #sin, cos, tan# are:

#sin2x=2sinxcosx#
#cos2x=cos^2x-sin^2x=1-2sin^2x=2cos^2x-1#
#tan2x=(2tanx)/(1-tan^2x)#.

Then, let's prove the given equation.

(LHS)#=(1-cos8x)/8=(1-cos(2*4x))/8#
#=(1-(1-2sin^2 4x))/8#
#=(2sin^2 4x)/8=(sin^2 4x)/4#

(RHS)#=sin^2 2x cos^2 2x#
#=(1/2*2sin2xcos2x)^2#
#=(1/2sin(2*2x))^2#
#=(1/2sin4x)^2=(sin^2 4x)/4#

In this way, (LHS)#=#(RHS) is proven.

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