The real roots of the equation `cos^7x+sin^4x=1` in the interval `(-pi,pi)` are __________, ________, and ________

Answer

`x= 0, pm pi/2`

Explanation

`cos^(7) x=1-sin^(4) x= (1-sin^(4) x)(1+sin^(2) x)`
or `cos^(7) x = cos^(2) x(2-cos^(2) x)`
or `cos^(2) x(cos^(5)x+cos^(2) x-2)=0`
or `cos x =0`
`rArr x=pm pi//2` in the given interval
or `cos^(5) x+cos^(2)x-2=0`,
which holds when `cos x=1`, hence `x=0`.
Thus, there are total three solutions : `0, pm pi//2`.