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# Prove that Tan {Pi/4 + 1by2 Cos^(-1) Abyb} + Tan {Piby4 - 1by2 Cos^(-1) Abyb} = (2b)Bya - CBSE (Commerce) Class 12 - Mathematics

ConceptProperties of Inverse Trigonometric Functions

#### Question

Prove that tan {pi/4 + 1/2 cos^(-1)  a/b} + tan {pi/4 - 1/2 cos^(-1)  a/b} = (2b)/a

#### Solution

Let cos^(-1) (a/b) = 0

Then cos theta = a/b

L.H.S:

tan {pi/4 + 1/4 cos^(-1)  a/b} + tan (pi/4 - 1/2cos^(-1)  a/b)

= tan (pi/4 + theta/2) + tan (pi/4 - theta/2)

= (1 + tan  theta/2)/(1 - tan  theta/2) + (1 - tan  theta/2)/(1 + tan  theta/2)

((1 + tan  theta/2)^2 + (1 - tan  theta/2)^2)/(1 - tan^2   theta/2)

= 2((1 + tan^2  theta/2)/(1 -  tan^2  theta/2))

= 2/(cos theta)   [∵ cos 2 theta = (1 - tan^2 theta)/(1 + tan^2 theta)]

= (2b)/a

=RHS

LHS RHS

Hence Proved

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Solution Prove that Tan {Pi/4 + 1by2 Cos^(-1) Abyb} + Tan {Piby4 - 1by2 Cos^(-1) Abyb} = (2b)Bya Concept: Properties of Inverse Trigonometric Functions.
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