Question

Sin A + sin cube A is equal to cos square A prove that cos 6 a minus 4 cos power 4 + 8 cos square A is equal to 4

Answer 1

It is given that:

$sinA+sin^3A=cos^2A$

Take common factor:

$sinA(1+sin^2A) = cos^2A$

From trigonometric rules: $sin^2=1-cos^2A$. Therefore,

$sinA(2 -cos^2A) = cos^2A$

Square both sides:

$sin^2A(4-4cos^2A +cos^4A) = cos^4A\\(1-cos^2A)(4-4cos^2A +cos^4A) = cos^4A\\4-4cos^2A +cos^4A-4cos^2A+4cos^4A-cos^6A = cos^4A\\4 -cos^6A +4cos^4A -8cos^2A = 0$

Hence, $cos^6A - 4 cos^4A + 8cos^2A = 4$