How do you prove #sin^2 (x) + cos^2 (x) = 1# using other trigonometric identities?

Typically, the given identity is just proven using the unit circle and the Pythagorean theorem, simple calculus, or a variety of other methods. Even if one proves it using other methods, it is important to remember that we cannot prove all trig identities only using other trig identities, as it would be relying on circular reasoning, and so even if we do prove it using another trig identity, we must use other methods somewhere along the line. It so happens that #sin^2(x) + cos^2(x) = 1# is one of the easier identities to prove using other methods, and so is generally done so.

Still, be all that as it may, let's do a proof using the angle addition formula for cosine:

#cos(alpha + beta) = cos(alpha)cos(beta) - sin(alpha)sin(beta)#

(A proof of the above formula may be found here )

Also, note that sine is an odd function and cosine is an even function, meaning

#sin(-x) = -sin(x)# and
#cos(-x) = cos(x)#

Now we proceed to the proof.

Let #alpha = x# and #beta = -x#

#=> cos(x + (-x)) = cos(x)cos(-x) - sin(x)sin(-x)#

#=> cos(0) = cos(x)cos(x) - (-sin(x)sin(x))#

#:. 1 = cos^2(x) + sin^2(x)#