Question

If cot theta +cos theta =p and cot theta - cos theta =q then prove tha p^2-q^2 =4√pq

Answer 1

cotФ + cosФ = p ------(1)
cotФ - cosФ = q ------(2)

LHS = P² - q²
= (cotФ + cosФ)² - (cotФ - cosФ)²
= cot²Ф + cos²Ф + 2cosФ.cotФ - cot²Ф - cos²Ф + 2cosФ.cotФ
= 4cotФ.cosФ

RHS = 4√pq
= 4√{(cotФ + cosФ)(cotФ - cosФ)}
= 4√{cot²Ф - cos²Ф}
= 4√{1/tan²Ф - 1/sec²Ф}
= 4√{(sec²Ф - tan²Ф)/tan²Ф.sec²Ф}
= 4√{1/tan²Ф.sec²Ф} [∵ from Identity sec²x -tan²x = 1 ]
= 4√{cot²Ф.cos²Ф}
= 4cotФ.cosФ

LHS = RHS [hence proved]