Question

If sec
$if \: \sec(a ) + \tan(a ) = p. \: then \: show \: that \: \csc(a)$
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Answer 1

Step-by-step explanation:

If sec

$if \: \sec(a ) + \tan(a ) = p. \: then \: show \: that \: \csc(a)$

Answer 2

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.