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

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