Trigonometry> if tan 2 a = 1 – e 2 then show that sec a...

if tan 2 a = 1 – e 2 then show thatsec a + tan 3 a cosec a = (2-e 2)3/2

Prateek Agrawal , 10 Years ago

Grade 12

2 Answers

Kaustubh Nayyar

Last Activity: 10 Years ago

tan²a=1- e² i.e. sec²a =2-e²

seca + tan³a coseca = 1/cosa + sin²a/cos³a = 1/cosa + (1-cos²a)/cos³a

= 1/cosa + 1/cos³a – 1/cosa = 1/cos³a = sec³a = (2-e²)^3/2

Rishi Sharma

Last Activity: 4 Years ago

Dear Student,
Please find below the solution to your problem.

tan²θ=1-a²
LHS
=secθ+tan³θcosecθ
=√(1+tan²θ)+tan²θ×tanθ×√(1+cot²θ)
[∵, sec²θ-tan²θ=1 and cosec²θ-cot²θ=1]
=√1+(1-a²)+(1-a²)×√(1-a²)×√{1+(1/tan²θ)}
=√(2-a²)+(1-a²)×√(1-a²)×√{1+1/(1-a²)}
=√(2-a²)+(1-a²)×√(1-a²)×√{(1-a²+1)/(1-a²)}
=√(2-a²)+(1-a²)×√(2-a²)
=√(2-a²)×(1+1-a²)
=√(2-a²)×(2-a²)
=(2-a²)¹/²⁺¹
=(2-a²)³/²
=RHS (Proved)

Thanks and Regards