Question

If sec theta + tan theta = p, show that sec theta - tan theta =1/p .hence,find the value of cos theta and sin theta

Answer 1

check the uploaded image

Answer 2

Secθ - Tanθ = 1/p  if Secθ  + Tanθ  = p

Step-by-step explanation:

Secθ  + Tanθ  = p

=> 1/Cosθ + Sinθ/Cosθ = p

=> (1 + Sinθ)/Cosθ = p

=> Cosθ/(1 + Sinθ)  = 1/p

=> Cosθ(1  - Sinθ) /(1 + Sinθ)(1 - Sinθ)  = 1/p

=> Cosθ(1 - Sinθ)/(1 - Sin²θ) = 1/p

=> Cosθ(1 - Sinθ)/Cos²θ= 1/p

=> (1 - Sinθ)/Cosθ= 1/p

=> 1/Cosθ - Sinθ/Cosθ = 1/p

=> Secθ - Tanθ = 1/p

(1 + Sinθ)/Cosθ = p

=> (1 + Sinθ) = pcosθ

Squaring both sides

=> 1 + Sin²θ + 2Sinθ = p²cos²θ

=>  1 + Sin²θ + 2Sinθ = p² - p²sin²θ

=>( p²+ 1)sin²θ  + 2Sinθ + (1 - p²) = 0

Sinθ = (- 2 ± √4 - 4(( p²+ 1)(1 - p²) ) / (2 ( p²+ 1))

= (-2  ± √4 -4 + 4p⁴)/ (2 ( p²+ 1))

= (-2 ± 2p²)/ 2(p²+ 1)

= -(1 ± p²)/(p²+ 1)

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