# Why does cos (90 - x) = sin (x) and sin (90 - x) = cos (x) ?

We need to find out the reason for cos (90-x)= sin x and (sin (90-x)=cos x

### Solution

We can get these by the cartesian quadrant graph or by trigonometric identity

We know the trigonometric identity that

$\cos (\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta$——————(i)

$\\sin (\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin \beta$—————–(ii)

Using equation (i)

$\cos (90-x )=\cos 90\cos x+\sin 90 \sin x$

On substituting cos 90 value as 0 and sin 90 as 1 we get,

$\cos (90-x )=0\cos x+1 \sin x$ $\cos (90-x )=\sin x$

Similarly, on solving (ii) we get

$\sin (90-x )=\cos x$

Hence proved