Question

Show that costheta/1-sintheta+1-sintheta/costheta=2sectheta

Answer 1

Answer:

$\frac{cosθ}{1 - sinθ} + \frac{1 - sinθ}{cosθ} = 2secθ \\ \frac{ {(cosθ)}^{2} + {(1 - sinθ)}^{2}}{cosθ(1 - sinθ)} = 2secθ \\ \frac{ {cos}^{2} θ + 1 - 2sinθ + {sin}^{2} θ}{cosθ(1 - sinθ)} = 2secθ \\ \frac{ {cos}^{2} θ + {sin}^{2}θ + 1 - 2sinθ}{cosθ(1 - sinθ)} = 2secθ \\ \frac{1 + 1 - 2sinθ}{cosθ(1 - sinθ)} = 2secθ \\ \frac{2 - 2sinθ}{cosθ(1 - sinθ)} = 2secθ \\ \frac{2(1 -sin θ)}{cosθ(1 - sinθ)} = 2secθ \\ \frac{2}{cosθ} = 2secθ \\ 2 \times \frac{1}{cosθ} = 2secθ \\ 2secθ = 2secθ \\ LHS=RHS \\ Hence \: proved$