Prove that (i) sin(pi+x)=-sin x (ii) sin(pi-x)=sin x (iii) cos(pi+x)=-cos x (ii) cos(pi-x)=-cos x

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In this tutorial you will learn the proof of the following 4 very important trigonometric identities:
(i) sin(pi+x) = -sin x
(ii) sin(pi-x) = sin x
(iii) cos(pi+x) = -cos x
(ii) cos(pi-x) = -cos x

These can also be written as
(i) sin(pi+theta) = -sin theta
(ii) sin(pi-theta) = sin theta
(iii) cos(pi+)theta = -cos theta
(ii) cos(pi-theta) = -cos theta

or
(i) sin(π+x) = -sin x
(ii) sin(π-x) = sin x
(iii) cos(π+x) = -cos x
(ii) cos(π-x) = -cos x

00:00 About it
00:13 cos(pi + x) = -cos x
01:14 cos(pi - x) = -cos x (Solution 1)
02:04 cos(pi - x) = -cos x (Solution 2)
03:21 sin(pi + x) = -sin x
04:08 sin(pi - x) = sin x

To prove these identities, you can use already proven identities of cos(x+y), cos(x-y), sin(x+y) and sin(x-y). The proof of identity cos(x+y) = cos x cos y - sin x sin y is one of the most important derivation. It is recommended to learn the proof / derivation of this identity in case you are not aware of it. Click the following link to watch the derivation:

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