Prove that $\frac{cos A - sin A + 1}{cos A + sin A - 1} = csc A + cot A$ $(cosA - sinA + 1)/(cosA + sinA - 1) = cscA + cotA$ ?

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Answer 1 · 2016-09-14T10:10:18

$L H S = \frac{cos A - sin A + 1}{cos A + sin A - 1}$ $LHS=(cosA-sinA+1)/(cosA+sinA-1)$

$= \frac{sin A (cos A - sin A + 1)}{sin A (cos A + sin A - 1)}$ $=(sinA(cosA-sinA+1))/(sinA(cosA+sinA-1))$

$= \frac{sin A cos A - {sin}^{2} A + sin A}{sin A (cos A + sin A - 1)}$ $=(sinAcosA-sin^2A+sinA)/(sinA(cosA+sinA-1))$

$= \frac{sin A cos A + sin A - (1 - {cos}^{2} A)}{sin A (cos A + sin A - 1)}$ $=(sinAcosA+sinA-(1-cos^2A))/(sinA(cosA+sinA-1))$

$= \frac{sin A (cos A + 1) - (1 - cos A) (1 + cos A)}{sin A (cos A + sin A - 1)}$ $=(sinA(cosA+1)-(1-cosA)(1+cosA))/(sinA(cosA+sinA-1))$

$= \frac{(1 + cos A) (sin A + cos A - 1)}{sin A (cos A + sin A - 1)}$ $=((1+cosA)(sinA+cosA-1))/(sinA(cosA+sinA-1))$

$=((1+cosA)cancel((sinA+cosA-1)))/(sinAcancel((cosA+sinA-1)))$

$=1/sinA+cosA/sinA$

$=cscA+cotA=RHS$

Proved

Prove that (cosA - sinA + 1)/(cosA + sinA - 1) = cscA + cotAcosA−sinA+1cosA+sinA−1=cscA+cotA(cosA - sinA + 1)/(cosA + sinA - 1) = cscA + cotA ?