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Question
Prove the following trigonometric identities.
sec A (1 − sin A) (sec A + tan A) = 1
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Solution
We have to prove sec A(1 − sin A)(sec A + tan A) = 1
We know that sec2 A − tan2 A − 1
So,
sec A(1 − sin A)(sec A + tan A) = {sec A(1 − sin A)}(sec A + tan A)
= (sec A − sec A sin A)(sec A + tan A)
= `(sec A - 1/cos A sin A) (sec A + tan A)` ...`(∵ sec theta = 1/costheta)`
= `(sec A - sin A/cos A) (sec A + tan A)` ...`(∵ tan theta = sin theta/costheta)`
= (sec A − tan A)(sec A + tan A)
= sec2 A − tan2 A
= 1 = R.H.S. ... (∵ sec2 θ = 1 tan2 θ)
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