Question

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

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Answer 1

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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

Answer:

Given , sec theta + tan theta=p------>1

=>sec theta - tan theta=1/p (as (sec theta + tan theta)

(sec theta- tan theta)=1)

------->2

1+2 => 2 sec theta=p^2+1/p----->3

1-2 => 2 tan theta=p^2-1/p------->4

Now do 3/4

=>cosec theta = p^2+1/p^2-1

I hope it helps you ✌