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Question
Prove the following identity:
`tantheta/(sectheta - 1) = (sectheta + 1)/tantheta`
Solution
L.H.S. = `(tan theta)/(sec theta - 1)`
= `(tan theta)/(sec theta - 1) xx (sec theta + 1)/(sec theta + 1)`
= `(tan theta (sec theta + 1))/(sec^2 theta - 1)` ...[a2 - b2 = (a + b)(a - b)]
= `(tan theta (sec theta + 1))/(tan^2 theta) ...[(1 + tan^2 theta = sec^2theta),(tan^2theta = sec^2theta - 1)]`
= `(cancel(tan theta) (sec theta + 1))/(cancel(tan^2 theta)_(tan theta))`
= `(sec theta + 1)/(tan theta)`
L.H.S. = R.H.S.
Hence proved.
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