If tan^2A= (1-e^2), show that secA+tan^3A= (2-e^2)^3/2.
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Answer:
Step-by-step explanation:
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)
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