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Question
Prove the following identities:
(1 + cot θ – cosec θ)(1 + tan θ + sec θ) = 2
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Solution
L.H.S. = (1 + cot θ – cosec θ)(1 + tan θ + sec θ)
= `(1 + cos theta/sin theta - 1/sin theta)(1 + sin theta/cos theta + 1/cos theta)`
= `((sin theta+ cos theta - 1)/sin theta)((cos theta + sin theta + 1)/cos theta)`
= `((sin theta + cos theta)^2 - 1^2)/(sin theta cos theta)` .....[∵ (a − b)(a + b) = a2 − b2]
= `(sin^2theta + cos^2theta + 2sintheta costheta - 1)/(sintheta costheta)`
= `(1 + 2sintheta costheta - 1)/(sin theta cos theta)`
= `(2sinthetacostheta)/(sintheta costheta)`
= 2
= R.H.S.
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