Question

If (cosec A + cot A) = m then prove that `(m^2 - 1)/(m^2 + 1) = cos A`.

Answer 1

Given: Let m = cosec A + cot A

To Prove: `(m^2 - 1)/(m^2 + 1) = cos A`

Proof [Step-wise]:

1. Note the identity cosec² A – cot² A = 1, so (cosec A + cot A)(cosec A – cot A) = 1.

2. Since cosec A + cot A = m, from step 1 we get `"cosec"  A − cot A = 1/m`.

3. Add the two linear equations:

`("cosec"  A + cot A) + ("cosec"  A − cot A) = m + 1/m` 

⇒ `2  "cosec"  A = m + 1/m` 

⇒ `"cosec"  A = (m^2 + 1)/(2m)`

4. Subtract the second from the first:

`("cosec"  A + cot A) - ("cosec"  A − cot A) = m - 1/m`

⇒ `2 cot A = m - 1/m` 

⇒ `cot A = (m^2 - 1)/(2m)`

5. Now express cos A as `(cot A)/("cosec"  A)`:

`cos A = (cot A)/("cosec"  A)` 

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

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

Therefore `(m^2 - 1)/(m^2 + 1) = cos A`, as required.