Simplify the expression (sin(a)cos(b)+cos(a)sin(b))/(cos(a)cos(b)-sin(a)sin(b)) * (cos(a)cos(b)+sin(a)sin(b))/(sin(a)cos(b)-cos(a)sin(b))?

2 Answers
Apr 27, 2017

tan(a+b) * cot(a-b)

Explanation:

The expression is:

E=(sin(a)cos(b)+cos(a)sin(b))/(cos(a)cos(b)-sin(a)sin(b)) * (cos(a)cos(b)+sin(a)sin(b))/(sin(a)cos(b)-cos(a)sin(b))

We can us the sine and cosine sum identities:

sin(A+B)=sinAcosB+cosAsinB
sin(A-B)=sinAcosB-cosAsinB
cos(A+B)=cosAcosB-sinAsinB
cos(A-B)=cosAcosB+sinAsinB

Applying these identities we can rewrite the expression as:

E = (sin(a+b))/(cos(a+b)) * (cos(a-b))/(sin(a-b))

\ \ \ = tan(a+b) * cot(a-b)

Apr 27, 2017

Tan (a+b)/ Tan (a-b)

Explanation:

To solve this, you need to know these formulae:

Sin (x+y) = Sin(x)Cos(y) + Cos(x)Sin(y)-----equation 1
Cos (x+y) = Cos(x)Cos(y) - Sin(x)Sin(y)----equation 2

If you replace y with -y in equation 1, we get,
Sin (x-y) = Sin(x)Cos(y) - Cos(x)Sin(y)

Similarly, replacing y with -y in equation 2, we get,
Cos (x-y) = Cos(x)Cos(y) + Sin(x)Sin(y)

Now let's move on to the question.
Simplifying all the terms, we get
(Sin (a+b))/Cos (a+b) xx (Cos (a-b))/(Sin (a-b))

Tan (a+b)/ Tan (a-b) is the answer