Question #058c3

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Answer 1 · 2018-02-17T04:11:09

$tan(a/2)$

$= sin(a/2)/cos(a/2)$ ---(1)

Using the identities

$sin(a/2) = sqrt[(1 - cosa)/2]$

And

$cos(a/2) = sqrt[(1 + cosa)/2]$

Put these values in (1)

We get , $tan(a/2)$

$= sqrt[(1 - cosa)/2]/ sqrt[(1 + cosa)/2]$

$= sqrt[(1 - cosa)/(1 + cosa)]$

Multiply both numerator and denominator by $1 - cosa$

$= sqrt[(1-cosa)/(1+cosa) times (1-cosa)/(1-cosa)]$

$= sqrt[(1-cosa)^2 /(1-cos^2a)]$

We know , $1 - cos^2a = sin^2a$

$= sqrt[(1-cosa)^2 / sin^2a$

$= sqrt[((1-cosa)/sina)^2]$

$= (1-cosa)/sina$

$= 1/sina - cosa/sina$

$= csc a - cota$