Question

If m cosec theta+n cot theta=2 and m^2 cosec^2 theta-n^2 cot^2 theta=5 then 81/m^2-1/n^2 is

Answer 1

It has given that m cosecθ + n cotθ = 2 and m²cosec²θ - n²cot²θ = 5

To find : The value of 81/m² - 1/n²

solution : here m²cosec²θ - n²cot²θ = 5

⇒(m cosecθ - n cotθ)(m cosecθ + n cotθ) = 5

⇒(m cosecθ - n cotθ) × 2 = 5 [as (m cosecθ + n cotθ) = 2]

⇒(m cosecθ - n cotθ) = 5/2 .........(1)

(m cosecθ + n cotθ) = 2.......(2) [ given ]

solving equations (1) and (2) we get,

2m cosecθ = 2 + 5/2 = 9/2

⇒m = 9/4 sinθ .........(3)

and 2n cotθ = 2 - 5/2 = -1/2

⇒ n = -1/4 tanθ .......(4)

now 81/m² - 1/n²

= 81/(9/4 sinθ)² - 1/(-1/4 tanθ)²

= 81/(81/16 sin²θ) - 1/(1/16 tan²θ)

= 16/sin²θ - 16/tan²θ

= 16[cosec²θ - cot²θ ]

we know, cosec²θ - cot²θ = 1

= 16 × 1

= 16

Therefore the value of 81/m² - 1/n² = 16