Prove :1-sinx/1+sinx=tan²(x/2 -pi/4)?

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Answer 1 · 2018-04-27T05:26:32

Please go through a Proof given in the Explanation.

I hope the Problem is to prove :

#(1-sinx)/(1+sinx)=tan^2(x/2-pi/4)#.

We know that, #sinx=(2tan(x/2))/(1+tan^2(x/2))#.

#:. 1/sinx=(1+tan^2(x/2))/(2tan(x/2))#.

Using dividendo componendo, we get,

#(1-sinx)/(1+sinx)=(1+tan^2(x/2)-2tan(x/2)}/(1+tan^2(x/2)+2tan(x/2)#,

#=(tan(x/2)-1)^2/(tan(x/2)+1)^2#,

#={(tan(x/2)-1)/(1+tan(x/2))}^2#,

#={(tan(x/2)-tan(pi/4))/(1+tan(x/2)*tan(pi/4))}^2#.

# rArr (1-sinx)/(1+sinx)={tan(x/2-pi/4)}^2=tan^2(x/2-pi/4)#,

as desired!

Enjoy Maths!