If cosec theta - sin theta = m and sec theta - cos theta = n find the relation between mand n by eliminating theta

We are given the equations:

cosecθ - sinθ = m
secθ - cosθ = n
We need to eliminate θ and find the relation between m and n.

Step 1: Express in Terms of Sin and Cos
Using trigonometric identities:

cosecθ = 1/sinθ
secθ = 1/cosθ
Rewriting the given equations:

(1/sinθ) - sinθ = m
⇒ (1 - sin²θ) / sinθ = m
⇒ cos²θ / sinθ = m …(i)

(1/cosθ) - cosθ = n
⇒ (1 - cos²θ) / cosθ = n
⇒ sin²θ / cosθ = n …(ii)

Step 2: Multiply Equations (i) and (ii)
Multiplying both sides:

(cos²θ / sinθ) * (sin²θ / cosθ) = m * n

Simplify:

(cos²θ * sin²θ) / (sinθ * cosθ) = m * n

Using sinθ * cosθ in the denominator:

(sinθ * cosθ) / (sinθ * cosθ) = m * n

Since (sinθ * cosθ) / (sinθ * cosθ) = 1,

We get:

m * n = 1

Conclusion:
The required relation between m and n is:

m * n = 1