Question

if tan theta + sec theta = x, show that sin theta = x²-1/x²+1

Answer 1

x² - 1 / x² + 1

= (secθ + tanθ)²- 1 / (secθ + tanθ)²+ 1

= sec²θ + tan²θ + 2secθtanθ - 1 / sec²θ + tan²θ - 2secθtanθ + 1

= (sec²θ - 1) + tan²θ + 2secθtanθ / (1 + tan²θ) + sec²θ + 2secθtanθ

= tan²θ + tan²θ + 2secθtanθ / sec²θ + sec²θ + 2secθtanθ

= 2tan²θ + 2secθtanθ / 2sec²θ + 2secθtanθ

= 2tanθ (tanθ + secθ) / 2secθ (tanθ+ secθ)

= tanθ / secθ

= sinθ / cosθ / 1 / cosθ

= sinθ / cosθ * cosθ

= sinθ

Therefore it is proved thatsinθ =x² - 1 / x² + 1.