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Question
Prove the following trigonometric identities.
(1 + cot A − cosec A) (1 + tan A + sec A) = 2
Solution
We have to prove (1 + cot A − cosec A) (1 + tan A + sec A) = 2
We know that, `sin^2 A + cos^2 A = 1`
So.
`(1 + cot A − cosec A) (1 + tan A + sec A) = (1 + cosA/sin A - 1/ sinA) (1 + sin A/cos A + 1/cos A)`
`= ((sin A + cos A - 1)/sin A)((cos A + sin A + 1)/cos A)`
`= ((sin A + cos A -1)(sin A + cos A + 1))/(sin A cos A)`
`= ({(sin A + cos A) - 1}{(sin A + cos A) + 1})/(sin A cos A)`
`= ((sin A + cos A)^2 -1)/(sin A cos A)`
`= (sin^2 A + 2 sin A cos A + cos^2 A - 1)/(sin A cos A)`
`= ((sin^2 A + cos^2 A) + 2 sin A cos A - 1)/(sin A cos A)`
`= (1 + 2 sin A cos A -1)/(sin A cos A)`
`= (2 sin A cos A)/(sin A cos A)`
= 2
Hence proved.
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