# A doubt regarding proof of values of trigonometric functions at allied angles

There are certain identities that help us to determine the values of trigonometric functions at $$\dfrac{\pi}{2}+x \text{, } \pi-x$$ etc. given the values of $$\sin x, \cos x$$.

Now, when we prove such identities, we usually take the value of $$x$$ to be in the interval $$\Big (0, \dfrac{\pi}{2} \Big )$$. Isn't it necessary to prove the identities by taking the value of $$x$$ in all $$4$$ quadrants individually and then arriving at the outcome? If not, then why not?

Pardon me if this sounds silly.

Thanks!

If the basis of your proofs are from Euler's identity $$e^{i \theta} = \cos \theta + i \sin \theta$$ then the quadrant becomes irrelevant.
Also consider the identities $$\cos \theta = \dfrac {e^{i \theta} + e^{-i \theta} } 2$$ and $$\sin \theta = \dfrac {e^{i \theta} - e^{-i \theta} } 2$$.