Question

For 0<x<pi/2, sin x and cos x are both less than 1 and greater than 0 (easy to see). We are also given that sin^2x+cos^2x=1. Use this to show that sin^7x+cos^7x<1 for 0<x<pi/2.

Unsure on how to proceed?

Answer 1

For the given domain,
since sinx < 1, sin^7x < sin^2x
same for cosx

so, sin^7 + cos^7 < sin^2 + cos^2 = 1

In general holds true for any power greater than 2.

Answer 2

Thanks man