Question

Prove this identity.
cos(3π/s +x) = sin x

Answer 1

If your question means:

cos ( 3 π / 2 + x ) = sin x

then

cos ( 3 π / 2 + x ) = cos ( 2 π / 2 + π / 2 + x ) =

cos [ π + ( π / 2 + x ) ]

Use identity:

cos ( π + θ ) = - cos θ

In this case:

cos [ π + ( π / 2 + x ) ] = - cos ( π / 2 + x )

Now

cos [ π + ( π / 2 + x ) ]

becomes

- cos ( π / 2 + x )

Now use identity:

cos ( π / 2 + θ ) = - sin θ

In this case:

cos ( π / 2 + x ) = - sin x

That's why:

- cos ( π / 2 + x ) = - ( - sin x ) = sin x

Answer 2

or
using cos(A+B) = cosAcosB - sinAsinB

cos(3π/2 + x) = cos 3π/2 cosx - sin 3π/2 sinx
= 0 cosx - (-1)sinx
= sinx