Question

Sin +cos = {sin cos}^2

Answer 1

here you didn't mention , angle of any trigonometry function .

we let x is angle of all trigonometry function ,

e.g sinx +cosx ={sinx .cosx}^2

take both sides square ,

(sinx + cosx )^2 = (sinx.cosx)^4

sin^2x + cos^2x +2sinx.cosx =(sinx.cosx)^4

1 +2sinx.cosx = (sinx.cosx)^4
use formula.
sin2x =2sinx.cosx

1 + sin2x = 1/16sin^4(2x)

16 + 16sin2x =sin^4(2x)

we see in RHS. is always positive term exist .
hence, LHS is always positive

16 + 16sin2x ≥ 0

1 + sin2x ≥ 0

sin2x ≥ -1

we know, sinx always greater then -1 but less then 1
for x €R this is possible .