Question

If f(x) = `sin^2x + sin^2(x + pi/3) + cosx cos(x + pi/3) and g(5/4) = 1`, then (gof)(x) is equal to: ______ 

Options

• `1/2`

• 0

• sin x

• 1

• – 1

• 2

• – 2

Answer 1

1

Explanation:

f(x) = `sin^2x + sin^2(x + pi/3) + cosx cos(x + pi/3)`

= `sin^2x + [sin(x + pi/3)]^2 + cosx[cosx cos  pi/3 - sinx sin  pi/3]`

= `sin^2x + [sinx cos  pi/3 + cosx sin  pi/3]^2 + cosx[1/2 cosx - sqrt3/2 sinx]`

= `sin^2x + [sinx/2 + sqrt3/2  cosx]^2 + cos^2x/2 - sqrt3/2 sinx cosx`

= `sin^2x + sin^2x/4 + 3/4 cos^2x + cos^2x/2 + sqrt3/2 sinx cosx - sqrt3/2 sinx cosx`

= `5/4(sin^2x + cos^2x) = 5/4`

(gof)(x) = g[f(x)] = `g(5/4) = 1`