2(sin^6A+cos^6A)-3(sin^4A+cos^4A)=-1?

1 Answer
Mar 19, 2018

Please see explanation section.

Explanation:

We know that,
color(blue)((1) .x^4+y^4=(x^2+y^2)^2-2x^2y^2)
color(red)((2) .x^6+y^6=(x^2+y^2)^3-3x^2y^2(x^2+y^2))
(3).sin^2theta+cos^2theta=1

Here, we use above formula color(red)((2)) and color(blue)((1)) respectively.

LHS=2(sin^6A+cos^6A)-3(sin^4A+cos^4A)

=2[color(red)((sin^2A+cos^2A)^3-3sin^2Acos^2A(sin^2A+cos^2A))]-3color(blue)([(sin^2A+cos^2A)^2-2sin^2Acos^2A)]
Using (3),we get

=2[(1)^3-3sin^2Acos^2A(1)]-3[(1)^2-2sin^2Acos^2A]

=2-6sin^2Acos^2A-3+6sin^2Acos^2A

=2-3

=-1=RHS