Question

Trigonometry - Verify Identity:

cos (x - y) / sin (x + y) = 1 + cotxcoty / cotx + coty

Answer 1

The required verification is given below.

Explanation:

To verify:
`(\cos (x-y))/(\sin (x+y))=(1 + \cot x\cot y)/(\cot x + \cot y)`

Consider,

Left hand side =
`(\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 =
`(\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 =
`(1 +(\cos x\cos y)/(\sin x\sin y) )/((\sin x\cos y+\cos x\sin y)/(\sin x\sin y ) )`

Now we have identity

`cot x = (\cos x)/(\sin x)`

and
`cot y =(\cos y)/(\sin y)` then

Left hand side =
`(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.