Question

if secant theta + tan theta is equal to P show that secant theta minus tan theta is equal to one upon P hence find the values of cos theta and sin theta​

Answer 1

cosθ = 2P/(P² + 1) and sinθ = (P² - 1)/(P² + 1)

it is given that,

secθ + tanθ = P......(1)

and we have to show that secθ - tanθ = 1/P

and also we have to find value of cosθ and sinθ.

we know from trigonometric identities, sec²x - tan²x = 1

so, sec²θ - tan²θ = 1

⇒(secθ - tanθ)(secθ + tanθ) = 1

⇒(secθ - tanθ) × P = 1 [ from equation (1), ]

⇒(secθ - tanθ) = 1/P [ hence proved ] .......(2)

from equations (1) and (2), we get

secθ = 1/2(P + 1/P) = (P² + 1)/2P

⇒cosθ = 1/secθ = 2P/(P² + 1)

and sinθ =√( 1 - cos²θ)

= √{1 - 4p²/(P² + 1)²}

= (P² - 1)/(P² + 1)

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