Question

Trigonometry - Verify Identity: cos (x - y) / sin (x + y) = 1 + cotxcoty / cotx + coty

Answer 1

Answer:

The required verification is given below.

Step-by-step explanation:

To verify: $\frac{\cos (x-y)}{\sin (x+y)}=\frac{1 + \cot x\cot y}{\cot x + \cot y}$

Consider,

Left hand side = $\frac{\cos (x-y)}{\sin (x+y)}$

Now by using Trigonometric identities,

$cos (A-B) = sin A.sin B + cos A.cos B \\sin (A+B) = sin A.cos B + cos A.sin B$

Now we get,

Left hand side = $\frac{\sin x\sin y + \cos x\cos y}{\sin x\cos y+\cos x\sin y}$

Now dividing Numerator and Denominator by $sin x.sin y$ we get

Left hand side = $\frac{1 +\frac{\cos x\cos y}{\sin x\sin y} }{\frac{\sin x\cos y+\cos x\sin y}{\sin x\sin y } }$

Now we have identity

$cot x = \frac{\cos x}{\sin x}$

and $cot y =\frac{\cos y}{\sin y}$ then

Left hand side = $\frac{1 +\cot x\cot y}{\cot x + \cot y}$

Which is equal to our Right hand side required identity

This is the way we have

Left hand side = Right hand side   Proved.