eliminate θ between cosecθ-sinθ=m and secθ-cosθ=n
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Given:
cosecθ-sinθ=m and secθ-cosθ=n
Solution: If m = cosecθ - sinθ = (1 - sin²θ)/sinθ = cos²θ/sinθ............eqn(1) and
n = secθ - cosθ = (1 - cos²θ)/cosθ = sin²θ/cosθ.................eqn(2)
Then, m/n = cos^3θ/sin^3θ = cot^3θ and n/m = sin^3θ/cos^3θ = tan^3θ.
Therefore,tan²θ = (m/n)^(2/3) and cot²θ = (n/m)^(2/3).
squaring equations (1) n (2) & use the fact that cosecθsinθ = secθcosθ = 1,
m² = cosec²θ + sin²θ - 2 = cot²θ + sin²θ - 1 ......using identity cosec²θ - 1 = cot²θ
&
n² = sec²θ + cos² - 2 = tan²θ + cos²θ - 1. ......using identity sec²θ - 1 = tan²θ
m² + n² + 2 = cot²θ + tan²θ + sin²θ + cos²θ = cot²θ + tan²θ + 1.
Subtracting 1 from both the sides,
m² + n² + 1 = (m/n)^(2/3) + (n/m)^(2/3).
cosecθ-sinθ=m and secθ-cosθ=n
Solution: If m = cosecθ - sinθ = (1 - sin²θ)/sinθ = cos²θ/sinθ............eqn(1) and
n = secθ - cosθ = (1 - cos²θ)/cosθ = sin²θ/cosθ.................eqn(2)
Then, m/n = cos^3θ/sin^3θ = cot^3θ and n/m = sin^3θ/cos^3θ = tan^3θ.
Therefore,tan²θ = (m/n)^(2/3) and cot²θ = (n/m)^(2/3).
squaring equations (1) n (2) & use the fact that cosecθsinθ = secθcosθ = 1,
m² = cosec²θ + sin²θ - 2 = cot²θ + sin²θ - 1 ......using identity cosec²θ - 1 = cot²θ
&
n² = sec²θ + cos² - 2 = tan²θ + cos²θ - 1. ......using identity sec²θ - 1 = tan²θ
m² + n² + 2 = cot²θ + tan²θ + sin²θ + cos²θ = cot²θ + tan²θ + 1.
Subtracting 1 from both the sides,
m² + n² + 1 = (m/n)^(2/3) + (n/m)^(2/3).
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