Establish the following identity: CosA/1-TanA + SinA/1-CotA = CosA-SinA

cosA/(1-tanA) + sinA/(1-cotA)

cosA/(1-tanA) + sinAtanA/(tanA-1)
(cosA - sinAtanA)/(1-tanA)
(cos^2A-sin^2A) / cosA(1-tanA)
(cos^2A-sin^2A)/(cosA-sinA)
cosA+sinA

sure you don't have a typo somewhere?

To establish the given identity:

CosA/(1-TanA) + SinA/(1-CotA) = CosA - SinA

We can start by converting tangent (TanA) and cotangent (CotA) to their reciprocal trigonometric functions:

TanA = SinA/CosA
CotA = CosA/SinA

Now, substitute the values of TanA and CotA in the original identity:

CosA/(1-(SinA/CosA)) + SinA/(1-(CosA/SinA))

We can simplify the denominators by multiplying them by the conjugates:

CosA/((CosA-SinA)/CosA) + SinA/((SinA-CosA)/SinA)

Simplifying further by multiplying the numerator and denominator of each fraction:

CosA/(CosA-SinA) * (CosA/CosA) + SinA/(SinA-CosA) * (SinA/SinA)

CosA * (CosA/CosA) + SinA * (SinA/SinA)

Simplifying further:

CosA + SinA

Therefore, the identity is established as CosA/(1-TanA) + SinA/(1-CotA) = CosA - SinA.