Question

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

Answer 1

Answer:

(p² + 1)/p² - 1

Step-by-step explanation:

Given: sec θ + tan θ = p   ----- (i)

We know that sec²θ - tan²θ = 1

⇒ (secθ + tanθ)(secθ - tanθ) = 1

⇒ (p)(secθ - tanθ) = 1

⇒ secθ - tanθ = (1/p)   ----- (ii)

On solving (i) & (ii), we get

⇒ secθ + tanθ + secθ - tanθ = p + 1/p

⇒ 2secθ = p² + 1/p

⇒ secθ = (p² + 1)/2p

⇒ cosθ = (1/secθ)

            = 2p/p² + 1

Sin²θ = 1 - cos²θ

         = 1 - (2p/p² + 1)²

         = 1 - (4p²)/p⁴ + 1 + 2p²

         = (p⁴ + 1 + 2p² - 4p²)/p⁴ + 1 + 2p

         = (p⁴ + 1 - 2p²)/p⁴ + 1 + 2p

         = (p² - 1)²/(p² + 1)²

sin θ= p² - 1/p² + 1.

Now,

We know that cosecθ = (1/sinθ)

⇒ (p² + 1)/p² - 1.

Hope it helps!

Answer 2

Answer......

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)