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.