Question

Find the period for the function: cos⁴ x

Answer 1

we know, cos²x = (1 + cos2x)/2

so, cos⁴x = (cos²x)²

= {(1 + cos2x)/2}²

= (1 + cos2x)²/4

= (1 + cos²2x + 2cos2x)/4

= 1/4 + cos²2x/4 + cos2x/2

= 1/4 + (1 + cos4x)/8 + cos2x/2

so, period of cos⁴x = period of 1/4 + (1 + cos4x)/8 + cos2x/2

we know, period of cosx = 2π

period of cos2x = 2π/2 = π

period of cos4x = 2π/4 = π/2

so, period of 1/4 + (1 + cos4x)/8 + cos2x/2 = LCM { period of cos2x , period of cos4x}

= LCM { π , π/2}

= π

hence, period of cos⁴x = π