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Show that Sec H-1(Sin θ) =Log Cot ( θ 2 ). - Applied Mathematics 1

Sum

Show that sec h-1(sin θ) =log cot (`theta/2` ).

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Solution

LHS = sec h-1(sin θ)
Let y = sec h-1(sin θ)
sec hy = sin θ
`1/sintheta= 1/(secℎy)`
cos hy = cosec θ
y = cos h-1(cosec θ)
but cos h-1x = log |𝑥+ `sqrt(x2−1)`|
∴ y = log |cosec θ+ `sqrt(cosec2 θ−1)`|
∴ y = log |𝑐𝑜𝑠𝑒𝑐 𝜃+cot𝜃|
= log |`1/sintheta + costheta/sintheta` |
= log |`(1+costheta)/sintheta`|
= log |`(2 cos^2  theta/2)/(2cos  theta/2 sin  theta/2)`|
= log |`(cos  theta/2)/(sin  theta/2)`|
= log cot (`theta/2`).
= RHS
∴ LHS = RHS
Hence proved

Concept: Logarithmic Functions
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