How do you simplify #cos(x+y)/sin(x-y)-cot(x-y)# to trigonometric functions of x and y?

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Answer 1 · 2015-12-29T13:50:56

We can start off by writing #cot(x-y)# in terms of #sin(x-y)# and #cos(x-y)# and take it from there. The step by step explanation is given below.

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

Remember #cot(A) = cos(A)/sin(A)#

Our expression becomes

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

We are subtracting two fractions whose denominators are same.

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

This stage if you are comfortable with sum and difference formula you can proceed simplifying them.

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

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

Now let us get back to our problem we had

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

#(-2sin(x)sin(y))/sin(x-y)# This should do for a simplification.