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Question
If sec θ + tan θ = p, obtain the values of sec θ, tan θ and sin θ in terms of p
Solution
Given sec θ + tan θ = p
We have sec2 θ – tan2 θ = 1
(sec θ + tan θ) (sec θ – tan θ) = 1
p(sec θ – tan θ) = 1
sec θ – tan θ = `1/"p"`
(sec θ – tan θ) + (sec θ – tan θ) = `"p" + 1/"p"`
2 sec θ = `(1 + "p"^2)/"p"`
sec θ = `(1 + "p"^2)/(2"p")`
(sec θ + tan θ) – (sec θ – tan θ) = `"p" - 1/"p"`
sec θ + tan θ – sec θ + tan θ = `("p"^2 - 1)/"p"`
2 tan θ = `("p"^2 - 1)/"p"`
tan θ = `("p"^2 - 1)/(2"p")`
`(sin theta)/(cos theta) = ("p"^2 - 1)/(2"p")`
`sin theta* sec theta = ("p"^2 - 1)/(2"p")`
`sin theta ((1 + "p"^2)/(2"p")) = ("p"^2 - 1)/(2"p")`
sin θ = `("p"^2 - 1)/(2"p") xx (2"p")/(1 + "p"^2)`
sin θ = `("p"^2 - 1)/("p"^2 + 1)`
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