# Answer to Question #11459 in Trigonometry for vishesh

Question #11459

(cosA-sinA+1) / (cosA+sinA-1) = cosecA+cotA

Expert's answer

Prove

(cosA-sinA+1) / (cosA+sinA-1) = cosecA+cotA

1 - cos^2 A = sin^2 A

=> (1 - cosA) * (1 + cosA) = sinA * sinA

=> (1 + cosA) / sinA = sinA / (1 - cosA)

=> (1 + cosA - sinA) / (sinA - 1 + cosA) = (1 + cosA) / sinA = 1/sinA + cosA/sinA

... [because a/b = c/d => (a-c)/(b-d) = a/b]

=> (cosA - sinA + 1) / (cosA + sinA - 1) = cosecA + cotA.

(cosA-sinA+1) / (cosA+sinA-1) = cosecA+cotA

1 - cos^2 A = sin^2 A

=> (1 - cosA) * (1 + cosA) = sinA * sinA

=> (1 + cosA) / sinA = sinA / (1 - cosA)

=> (1 + cosA - sinA) / (sinA - 1 + cosA) = (1 + cosA) / sinA = 1/sinA + cosA/sinA

... [because a/b = c/d => (a-c)/(b-d) = a/b]

=> (cosA - sinA + 1) / (cosA + sinA - 1) = cosecA + cotA.

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