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Prove that: (1+cotA+cosecA)(1+cotA-cosecA) =cot(A/2)-tan(A/2)

Prove that:
(1+cotA+cosecA)(1+cotA-cosecA)
=cot(A/2)-tan(A/2)

Grade:11

1 Answers

Faiz
107 Points
4 years ago
Using this (a-b)(a+b) = a² - b².Where a=1+cota and b=coseca.-> (1+cota)² - cosec²a-> 1 + cot²a + 2cota - cosec²aIdentity: cosec²x - cot²x = 1 using it.-> 1 + (-1) + 2cota-> 2 cot a-> 2 cos a / sin aUsing half angle formula for cosine and sine.:-> [2 (cos² a/2 - sin²a/2)] / (2 sin a/2 cos a/2)-> (cos² a/2 - sin²a/2) / (sin a/2 cos a/2)Divide numerator and denominator.-> cos² a/2 / (sin a/2 cos a/2) - sin²a/2 / (sin a/2 cos a/2)-> cot a/2 - tan a/2..hence proved

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