trig identities (1 Viewer)

[jemsta]

can anyone help me prove this
cos@(tan@-tan1/2@)=tan1/2@ using the t substitiution method?

 

[Riviet]

Let t=tan(@/2)
then LHS=cos@(tan@-tan(@/2))
=(1-t²)/(1+t²) x (2t/(1-t²)-t)
=2t/(1+t²) - (t(1-t²)/(1+t²)
=(t+t³)/(1+t²)
=t(1+t²)/(1+t²)
=t
=tan(@/2)
There ya go ^^

 

[jemsta]

haha waiting

 

[Trebla]

Let t = tan(θ/2)
LHS = cosθ(tanθ - tan(θ/2))
= (1 - t²)/(1 + t²) [ (2t/1 + t²) - t]
Expand and simplify:
= 2t/(1 - t²) - t(1 - t²)/(1 + t²)
Place on common demonimator
= (2t - t + t³)/(1 + t²)
= (t + t³)/(1 + t²)
Factorise 't' in numerator
= t(1 + t²)/(1 + t²)
Eliminate/cross out (1 + t²)
= t
= tan(θ/2)
= RHS

 

[Trebla]

damn, got beaten by 1 minute.....although you saw the question first.......

 

[Riviet]

I took my time too xD