Are the given trigonometric identities correct?

[louie3006]

Homework Statement

tanX+tan Y / 1-tanXtanY= cotX+cotY/cotXcotY-1

Homework Equations

tan= sin/cos
cot= cos/sin

The Attempt at a Solution

 

[joeyar]

Attempt at a solution?

Try substituting those equations into your problem. Where does that get you?

 

[Architeuthis Dux]

To verify the given identity, we can rewrite both sides using the definitions of tangent and cotangent:

Left side:
tanX+tanY / 1-tanXtanY
= (sinX/cosX) + (sinY/cosY) / 1 - (sinX/cosX)(sinY/cosY)
= (sinXcosY + sinYcosX) / (cosXcosY - sinXsinY)

Right side:
cotX+cotY / cotXcotY-1
= (cosX/sinX) + (cosY/sinY) / (cosXcosY/sinXsinY) - 1
= (cosXsinY + cosYsinX) / (cosXcosY - sinXsinY)

Since both sides are equal, the given identity is verified.