Question

Prove that `cos A / (1- tan A) + sin A / (1- cot A) -= cos A + sin A`?

Answer 1

We want to prove that:

`cos A / (1- tan A) + sin A / (1- cot A) -= cos A + sin A`

If we manipulate the LHS we have:

`LHS = cos A / (1- tan A) + sin A / (1- cot A)`

`" " = cos A / (1- sinA/cosA) + sin A / (1- cosA/sinA)`

`" " = cos A / ((cosA- sinA)/cosA) + sin A / ((sinA- cosA)/sinA)`

`" " = cos^2 A / (cosA- sinA) + sin^2 A / (sinA- cosA)`

`" " = cos^2 A / (cosA- sinA) - sin^2 A / (cosA- sinA)`

`" " = (cos^2 A-sin^2A) / (cosA- sinA)`

`" " = ((cosA+sinA)(cosA-sinA)) / (cosA- sinA)`

`" " = cosA+sinA \ \ \` QED

Answer 2

We want to prove that

`cosa/(1-tana)+sina/(1-cota)=cosa+sina`

Rewrite the left using only sines and cosines:

`cosa/(1-sina/cosa)+sina/(1-cosa/sina)=cos^2a/(cosa-sina)+sin^2a/(sina-cosa)`

Pulling a `-1` out of the denominator of the second term gives us the same denominator:

`=cos^2a/(cosa-sina)-sin^2a/(cosa-sina)=(cos^2a-sin^2a)/(cosa-sina)`

Factoring:

`=((cosa+sina)(cosa-sina))/(cosa-sina)=cosa+sina`

Which was originally the right hand side, so this is proven.