Question

If cot 0 + cos 0 = m and cot 0 - cos 0 = n, then prove that (m2 - 12)2 = 16 mn.​

Answer 1

cotθ + cosθ = m

cotθ - cosθ = n

now,

(m² - n²)² = [(m - n)(m + n)]²

[ (cotθ + cosθ- cotθ + cosθ)(cotθ - cosθ+ cotθ + cosθ) ]²

[ (2cosθ)(2cotθ) ] ²

[ 16(cot²θ) . (cos²θ) ]-----------( 1 )

now,

16mn

16(cot θ + cos θ)(cot θ - cos θ)

16(cot²θ - cos²θ)

16(cos²θ / sin²θa - cos²θ)

16(cos²θ - cos²θ . sin²θ)/sin²θ

16(cos²θ(1 - sin²θ)/sin²θ

16(cos²θ . cos²θ)/sin²θ

16(cos²θ/sin²θ . cos²θ)

16(cot²θ . cos²θ)------------( 2 )

from-----( 1 ) & -----( 2 )

[(m² - n²)]² = 16(m n)

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