Question

Show that `sin^7 x + cos^7 x < 1` if `0 < x < frac{\pi}{2}`. Please explain in words how to calculate this?

Answer 1

When `x` is a non zero acute angle, both `sin x` and `cos x` are positive numbers smaller than one. For such numbers, higher powers are smaller than lower ones (since each time you raise the power you are multiplying by a number smaller than one). Thus `sin^7x` and `cos^7x` are smaller than `sin^2x` and `cos^2x`, respectively, since the latter add up to 1, the sum of the seventh powers must be smaller than 1.