Question

Prove that:- 2(sin 6 ө cos 6 ө)-3(sin 4 ө cos 4 ө) 1 =0

Answer 1

2(sin⁶θ+cos⁶θ)-3(sin⁴θ+cos⁴θ)+1
=2{(sin²θ)³+(cos²θ)³}-3{(sin²θ)²+(cos²θ)²}+1
=2(sin²θ+cos²θ)(sin⁴θ-sin²θcos²θ+cos⁴θ)-3{(sin²θ+cos²θ)²-2sin²θcos²θ}+1
=2(1)(sin⁴θ-sin²θcos²θ+cos⁴θ)-3{(1)²-2sin²θcos²θ}+1
[∵, sin²θ+cos²θ=1]
=2sin⁴θ-2sin²θcos²θ+2cos⁴θ-3+6sin²θcos²θ+1
=2sin⁴θ+4sin²θcos²θ+2cos⁴θ-2
=2{(sin²θ)²+2sin²θcos²θ+(cos²θ)²}-2
=2(sin²θ+cos²θ)²-2
=2(1)²-2
=2-2
=0 (Proved)