Question #efceb

1 Answer
NickTheTurtle
Jan 1, 2018

Start from the left-hand side:
#sin^2(x)cos^4(x)#
#=(1-cos^2(x))cos^4(x)# {since #sin^2(x)+cos^2(x)=1#}
#=cos^4(x)-cos^6(x)#

Compare it to the right-hand:
#cos^2(x)+cos^4(x)-cos^6(x)#

Thus, the equation in the question only holds when #cos^2(x)=0#, or #x=(2k+1)pi,kinZZ#. Any value outside this range invalidates the equation.

This can be verified by a graph of the two sides of the equation:
graph{(y-(sin(x))^2 (cos(x))^4)(y-(cos(x))^2-(cos(x))^4+(cos(x))^6)=0 [-10, 10, -0.25, 1.25]}

As seen, they only intersect when #x=(2k+1)pi,kinZZ#.

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