prove cofunction identity using addition and subtraction cos

prove cofunction identity using addition and subtraction

cos(x+ pi/3) + sin(x- pi/6) =0

Solution

We have sum and difference formula for sine and cosine

cos(A + B) = cos A cos B sin A sin B

sin(A B) = sin A cos B cos A sin B

Using these two formulae we have,

cos(x+pi/3) + sin(x- pi/6)

= cosx cospi/3 - sinx sin pi/3 + sinx cos pi/6 - cosx sin pi/6

= (1/2) cosx - (sqrt3/2) sinx + sqrt(3) /2 sinx - (1/2) cosx

= 0