Proving Trigonometric Identities?

Question

#cos(x-y)sinx-sin(x-y)cosx=siny#

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Answer 1 · 2018-06-21T20:41:27

See below

Use the following identities:

#cos(x-y)=sin(x) sin(y) + cos(x) cos(y)#
#sin(x-y)=sin(x) cos(y) - cos(x) sin(y)#

The expression becomes

#(sin(x) sin(y) + cos(x) cos(y))sin(x) - (sin(x) cos(y) - cos(x) sin(y))cos(x)#

#=sin^2(x) sin(y) + cancel(cos(x) cos(y) sin(x)) - (cancel(sin(x) cos(y) cos(x)) - cos^2(x) sin(y))#

#= sin^2(x) sin(y) + sin(y) cos^2(x) = sin(y)(sin^2(x)+cos^2(x))#

And since #sin^2(x)+cos^2(x)=1#, the result is proven