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Question
Prove the following trigonometric identities.
(secθ + cosθ) (secθ − cosθ) = tan2θ + sin2θ
Solution
We have to prove`(sec theta + cos theta)(sec theta - cos theta) = tan^2 theta + sin^2 theta`
We know that
`sin^2 theta + cos^2 theta = 1`
`sec^2 theta - tan^2 theta = 1`
`(sec theta + cos theta)(sec theta - cos theta) = sec^2 theta - cos^2 theta`
`= (1 + tan^2 theta) - (1 - sin^2 theta)`
`= 1 + tan^2 theta - 1 + sin^2 theta`
`= tan^2 theta + sin^2 theta`
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