Write the expression in terms of first powers of cosine cos4

Write the expression in terms of first powers of cosine. cos^4 x + sin^2 x

Solution

cos^4x +sin^2x

cos^4x = cos^2x (1-sin^2x)

cos^4x +sin^2x = cos^2x (1-sin^2x)+sin^2x

= cos^2x +sin^2x - cos^2xsin^2x

= 1 -(cosxsinx)

Use the formula sin2x = 2sinxcosx

So = 1 - (sin^2x)/4

    = (4- sin^2x)/4

Use the formula cos2x = 1-2sin^2x

So, = (4- sin^2x)/4

= (4 - (1-cos2x))/4

= (3+cos2x)/4