Question

Trigonometry - Verify Identity:

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

Answer 1

`(cos(x-y))/(sin(x+y))=(1+cot(x)cot(y))/(cot(x)+cot(y))`

`Left\ side = Right\ side`, hence the identity is verified.

Explanation:

`(cos(x-y))/(sin(x+y)=(1+cot(x)cot(y))/(cot(x)+cot(y))`

Working on right hand side.

`(1+cot(x)cot(y))/(cot(x)+cot(y))`

Substituting
`[cot(x)=(cos(x))/(sin(x))]` and
`[cot(y)=(cos(y))/(sin(y))]`

`=(1+(cos(x)cos(y))/(sin(x)sin(y)))/((cos(x))/(sin(x))+(cos(y))/(sin(y)))`

Taking LCD and adding fractions.

`=((sin(x)sin(y)+cos(x)cos(y))/(sin(x)sin(y)))/((cos(x)sin(y)+sin(x)cos(y))/(sin(x)sin(y)))`

Cancelling out the common denominators.

`=(sin(x)sin(y)+cos(x)cos(y))/(cos(x)sin(y)+sin(x)cos(y))}`

Applying sum and difference formulas
`[cos(x-y)=cos(x)cos(y)-sin(x)sin(y)][sin(x+y)=sin(x)cos(y)+sin(y)cos(x)]`

`=(cos(x-y))/(sin(x+y))`

Left side

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

∵
`Left\ side = Right\ side`, hence the identity is verified.