Question

if sec theta + tan theta = p, then find the value of cosec theta

Answer 1

secθ+tanθ=p                                                    ----------------------(1)

∵ sec²θ-tan²θ=1

or, (secθ+tanθ)(secθ-tanθ)=1

or, secθ-tanθ=1/p                                            ----------------------(2)

Adding (1) and (2) we get,

2secθ=p+1/p

or, secθ=(p²+1)/2p

∴ cosθ=1/secθ=2p/(p²+1)

∴ sinθ=√(1-cos²θ)

=√[1-{2p/(p²+1)}²]

=√[1-4p²/(p²+1)²]

=√[{(p²+1)²-4p²}/(p²+1)²]

=√[(p⁴+2p²+1-4p²)/(p²+1)²]

=√(p⁴-2p²+1)/(p²+1)

=√(p²-1)²/(p²+1)

=(p²-1)/(p²+1)

∴, cosecθ=1/sinθ=1/[(p²-1)/(p²+1)]= (p²+1)/(p²-1)

Answer 2

hey mate....
secQ+tanQ=p
secQ+sinQ/cosQ=p
secQ×cosQ+sinQ/cosQ=p
1/cosQ×cosQ+sinQ=p
1+sinQ=p cosQ
sinQ=p cosQ-1
1/sinQ=1/p cosQ-1
cosecQ=p cosQ+1/p^2cos^2Q-1