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