Question

If cos A = `(2sqrt("m"))/("m" + 1)`, then prove that cosec A = `("m" + 1)/("m" - 1)`

Answer 1

cos A = `(2sqrt("m"))/("m" + 1)`   ......[Given]

We know that,

sin²A + cos²A = 1

∴ `sin^2"A" + ((2sqrt("m"))/("m" + 1))^2` = 1

∴ `sin^2"A" + (4"m")/("m" + 1)^2` = 1

∴ sin²A = `1 - (4"m")/("m" + 1)^2`

= `(("m" + 1)^2 - 4"m")/("m" + 1)^2`

= `("m"^2 + 2"m" + 1 - 4"m")/("m" + 1)^2` ......[∵ (a + b)² = a² + 2ab + b²]

= `("m"^2 - 2"m" + 1)/("m" + 1)^2`

∴ sin²A = `("m" - 1)^2/("m" + 1)^2`   ......[∵ a² – 2ab + b² = (a – b)²]

∴ sin A = `("m" - 1)/("m" + 1)`  .....[Taking square root of both sides]

Now, cosec A =  `1/"sin A"`

= `1/(("m" - 1)/("m" + 1))`

∴ cosec A = `("m" + 1)/("m" - 1)`