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Prove tan(45+a)+tan(45-a)/tan(45+a)-tan(45-a)=cosec2a

Prove tan(45+a)+tan(45-a)/tan(45+a)-tan(45-a)=cosec2a

Grade:11

1 Answers

Arun
25750 Points
5 years ago
tan(45+A)+tan(45-A) = 

= [tan(45)+tanA] / [1-tan45.tanA] + [tan(45)-tanA] / [1+tan45.tanA] = 

= (1+tanA) / (1-tanA) + (1-tanA) / (1 + tanA) = 

= (1+tanA)^2 / (1-tan^2A) + (1-tanA)^2 / (1 - tan^2A) = 

= [(1+tanA)^2 + (1-tanA)^2] / (1-tan^2A) = 2(1+tan^2A) / (1-tan^2A) = 

= 2sec2A
 
similarly
 
tan(45+A) – tan(45-A) = 2 tan2A
hence answer will be = 2 sec2A / 2 tan2A = 1/ sin 2A = cosec 2A

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