Is secant theta + tan theta equal to p then find out the value of cosecant theta?
Answers
Answer:
sec theta + tan theta = p
=> (1 + sin theta)/cos theta = p
=> 1 + sin theta = p cos theta
square on both side
=> 1 + sin^2 theta + 2sin theta = p^2 cos^2 theta
=> 1 + sin^2 theta + 2 sin theta = p^2 ( 1 - sin^2 theta )
=> 1 + sin^2 theta + 2 sin theta = p^2 ( 1 - 1/cosec^2 theta )
=> 1 + sin^2 theta + 2 sin theta = p^2 ( cosec^2 theta - 1 )/cosec^2 theta
Multiplying whole equation by cosec^2 theta
=> cosec^2 theta + 1 + 2 cosec theta = p^2 ( cosec^2 theta - 1) .........(i)
since: (cosec theta + 1)^2 = cosec^2 theta + 1 + 2 cosec theta ............(ii)
Substituting (ii) in (i)
=> (cosec theta + 1)^2 = p^2(cosec^2 theta - 1)
=> (cosec theta + 1)^2/(cosec^2 theta -1) = p^2
=> (cosec theta + 1)(cosec theta + 1)/(cosec theta + 1)(cosec theta - 1) = p^2
=> (cosec theta + 1)/(cosec theta - 1) = p^2
=> cosec theta + 1 = p^2 cosec theta - p^2
=> p^2 + 1 = p^2 cosec theta - cosec theta
=> p^2 + 1 = cosec theta ( p^2 - 1 )
=> (p^2 + 1)/(p^2 - 1) = cosec theta
=> cosec theta = (p^2 + 1)/(p^2 - 1)
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