Question:
tan*-1{ cosx/ 1- sin x } , express it into simplest form
Answer:
Given, tan-1 {cos x/(1- sin x)}
= tan-1 {(cos2 x/2 - sin2 x/2)/(cos2 x/2 + sin2 x/2 - 2 * sin x/2 * cos x/2)}
= tan-1 {(cos2 x/2 - sin2 x/2)/(cos x/2 - sin x/2)2 }
= tan-1 [{(cos x/2 - sin x/2) * (cos x/2 + sin x/2)}/(cos x/2 - sin x/2)2 }]
= tan-1 {(cos x/2 + sin x/2)/(cos x/2 - sin x/2)}
= tan-1 [{(1 + (sin x/2)/ (cos x/2)}/{(1 - (sin x/2)/ (cos x/2)}]
= tan-1 {(1 + tan x/2)/(1 - tan x/2)}
= tan-1 {(tan π/4 + tan x/2)/(1 - tan π/4 * tan x/2)}
= tan-1 {(tan (π/4 + x/2)}
= π/4 + x/2
So, tan-1 {cos x/(1- sin x)} = π/4 + x/2
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