Question

If cotx(1 + sinx) = 4m and cotx(1 – sinx) = 4n, prove that (m² – n²)² = mn.

Answer 1

Given 4m = cotx (1+ sinx) and 4n = cotx (1 – sinx)

Multiplying both equations, we get

⇒ 16mn = cot²x (1 – sin²x)

We know that 1 – sin²x = cos²x

⇒ 16mn = cot²x cos²x

⇒ mn = \(\cfrac{cos^4\text x}{16\,sin^2\text x}\)....(1)

Squaring the given equations and then subtracting,

⇒ 16m² = cot²x (1+ sinx)² and 16n² = cot²x (1 – sinx)²

⇒ 16m² – 16n² = cot²x (4 sinx)

Squaring both sides,

From (1) and (2),

⇒ (m² – n²)

= mn

Hence proved.