Question

if sec theta + tan theta = P than find the value of cosec theta

Answer 1

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Step-by-step explanation:

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) Ans.

Answer 2

X = theta

sec X = 1/cosx

tanX = sinx/cosx

1/cosx + sinx/cosx = P

=>1 + sinx / cosx = p

=>(1+sinx)²/cos²X = p²

=>(1+sinx)²/1-sin²X = p²

=>(1+sinx)²/(1+sinx)(1-sinx) = p²

=>(1+sinx)/1-sinx= p²

=>cosecx+1/cosecx-1 =p²

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