Answer:
[tex]\frac{cos(x-y)}{sin(x+y)}=\frac{1+cot(x)cot(y)}{cot(x)+cot(y)}[/tex]
[tex]Left\ side = Right\ side[/tex], hence the identity is verified.
Step-by-step explanation:
[tex]\frac{cos(x-y)}{sin(x+y}=\frac{1+cot(x)cot(y)}{cot(x)+cot(y)}[/tex]
Working on right hand side.
[tex]\frac{1+cot(x)cot(y)}{cot(x)+cot(y)}[/tex]
Substituting [tex][cot(x)=\frac{cos(x)}{sin(x)}][/tex] and [tex][cot(y)=\frac{cos(y)}{sin(y)}][/tex]
[tex]=\frac{1+\frac{cos(x)cos(y)}{sin(x)sin(y)}}{\frac{cos(x)}{sin(x)}+\frac{cos(y)}{sin(y)}}[/tex]
Taking LCD and adding fractions.
[tex]=\frac{\frac{sin(x)sin(y)+cos(x)cos(y)}{sin(x)sin(y)}}{\frac{cos(x)sin(y)+sin(x)cos(y)}{sin(x)sin(y)}}[/tex]
Cancelling out the common denominators.
[tex]=\frac{sin(x)sin(y)+cos(x)cos(y)}{cos(x)sin(y)+sin(x)cos(y)}}[/tex]
Applying sum and difference formulas [tex][cos(x-y)=cos(x)cos(y)-sin(x)sin(y)][sin(x+y)=sin(x)cos(y)+sin(y)cos(x)][/tex]
[tex]=\frac{cos(x-y)}{sin(x+y)}[/tex]
Left side
[tex]\frac{cos(x-y)}{sin(x+y)}[/tex]
∵ [tex]Left\ side = Right\ side[/tex], hence the identity is verified.
