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Question
Prove that `sqrt(sec^2 theta + cosec^2 theta) = tan theta + cot theta`
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Solution
LHS:
`sqrt(sec^2theta + cosec^2theta)`
`= sqrt(1/cos^2theta + 1/sin^2theta)`
Taking LCM:
`= sqrt ((sin^2theta + cos^2theta)/(sin^2 theta cos^2theta))`
Using the identity sin2θ + cos2θ = 1:
`= sqrt(1/(sin^2theta cos^2theta))`
`= 1/(sintheta costheta)`
RHS:
tanθ + cotθ
`= sintheta/costheta + costheta/sintheta`
Taking LCM:
`= (sin^2theta + cos^2theta)/(sintheta costheta)`
Using sin2θ + cos2θ = 1:
`= 1/(sintheta costheta)`
LHS = RHS
`sqrt(sec^2theta + cosec^2theta) = tantheta + cottheta`
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