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

∴ `sin^2A + (4m)/(m + 1)^2 = 1`

∴ `sin^2A = 1 - (4m)/(m + 1)^2`

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

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

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

∴ `sin^2A = (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)`