How do you prove the identity $(tany+siny)/(2tantheta)=cos^2(y/2)$ ?

Question

2101 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-10-06T17:48:14

see below

$(tan y+siny)/(2tany) = cos^2 (1/2 y)$

Left Side: $=(tan y+siny)/(2tany)$

$=(siny/cosy+siny)/(2siny/cosy)$

$=((siny+sinycosy)/cosy)/(2siny/cosy)$

$=(siny+sinycosy)/cosy * cosy/(2siny)$

$=(siny+sinycosy)/(2siny)$

$=(siny(1+cosy))/(2siny)$

$=1/2 (1+cos y)$

$=(sqrt(1/2 (1+cos y)))^2$

$=(cos (1/2 y))^2$

$=cos^2(1/2 y)$

$=$ Right Side

How do you prove the identity (tany+siny)/(2tantheta)=cos^2(y/2)(tany+siny)/(2tantheta)=cos^2(y/2)?