SOLUTION: If {{{ tan^2 (45+theta)= a/b }}} prove that {{{ (b-a)/(b+a) = -sin2theta }}}

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Question 940703: If +tan%5E2+%2845%2Btheta%29=+a%2Fb+ prove that +%28b-a%29%2F%28b%2Ba%29+=+-sin2theta+
Answer by krutarthas(22) About Me  (Show Source):
You can put this solution on YOUR website!
+tan%5E2+%2845%2Btheta%29=+a%2Fb+
a/b =tanē (45+θ)
={tan (45+θ)}ē
={(tan 45+tan θ)/(1-tan 45.tan θ)}ē
={(1+tan θ)/(1-1.tan θ)}ē
=(1+tan θ)ē/(1-tan θ)ē
=(1+tanēθ+2tanθ)/(1+tanēθ-2tanθ)
(b-a)/(b+a)=((1+tanēθ-2tanθ)-(1+tanēθ+2tanθ))/((1+tanēθ-2tanθ)+(1+tanēθ+2tanθ))
=(1+tanēθ-2tanθ-1-tanēθ-2tanθ)/(1+tanēθ-2tanθ+1+tanēθ+2tanθ)
=4tanθ/2(1+tanēθ)
=2tanθ/secēθ
=2(sinθ/cosθ)/(1/cosēθ)
=2sinθ.cosēθ/cosθ
=2sinθ.cosθ
=sin2θ