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Question
Prove the following identities:
`1/(sectheta + tantheta) - 1/costheta = 1/costheta - 1/(sectheta - tantheta)`
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Solution
We have to prove that,
`1/(sectheta + tantheta) - 1/costheta = 1/costheta - 1/(sectheta - tantheta)`
`1/(sectheta + tantheta) + 1/(sectheta - tantheta) = 1/costheta + 1/costheta`
`1/(sectheta + tantheta) + 1/(sec theta - tan theta) = 2/costheta`
L.H.S. = `1/(sectheta + tantheta) + 1/(sec theta - tan theta)`
= `(sectheta - tantheta + sectheta + tantheta)/((sectheta + tantheta)(sectheta - tantheta))`
= `(2sectheta)/(sec^2theta - tan^2theta)` .....[a2 − b2 = (a + b) (a − b)]
= `(2secθ)/1` ...`[(because sec^2theta = 1 + tan^2theta),(therefore sec^2 theta - tan^2theta = 1)]`
= 2secθ
= `2 xx 1/(cosθ)`
= `2/costheta`
= R.H.S.
Hence proved.
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