Prove the following is an identity...

Question

Prove that  is an identity. 

Pleae keep formatting and answer neat and concise thank you!

Answer 1

-sinx -cosxcotx = -cscx

-sinx - cosx(cosx/sinx) = -1/sinx

multiply both sides by sinx

-sin^2(x) - cos^2(x) = -1

multiply both sides by -1

sin^2(x) + cos^2(x) = 1

1 = 1

sin^2(x) + cos^2(x) =1 is a basic trig identity and

is a version of the pythagorean theorem

on a unit circle, x^2 +y^2 = r^2 is a circle's equation,

with r=1, x =rcosx, y=rsinx

converting to polar coordinates

r^2(cos^2(x)) + r^2(sin^2(x)) = r^2

divide both sides by r^2

cos^2(x) + sin^2(x) = 1