Question

if sec theta+tan theta = p,than prove that sin theta= p²-1/p²+1​​​

Answer 1

Solution:

sec θ + tan θ = p

Squaring on both sides,

(sec θ + tan θ)² = p²

→ sec²θ + tan²θ + 2sec θ tan θ = p²

Substitute the value of p² in the given expansion

$\sf{ \dfrac{ {p}^{2} - 1}{ {p}^{2} + 1 } }$

$= \sf{\dfrac{sec^{2} \theta + tan^{2} \theta + 2sec \theta \: tan \theta - 1}{sec^{2} \theta + tan^{2} \theta + 2sec \theta \: tan \theta + 1}}$

Substitute

• tan²θ + 1 as sec²θ and
• sec²θ - 1 as tan²θ

$\implies \sf{\dfrac{tan^{2} \theta + tan^{2} \theta + 2sec \theta \: tan \theta}{ {sec}^{2} \theta + sec^{2} \theta + 2sec \theta \: tan \theta }}$

$\implies \sf{\dfrac{2 {tan}^{2} \theta + 2sec \theta tan \theta}{2 {sec}^{2} \theta + 2sec \theta tan \theta}}$

$\implies \sf{ \dfrac{ \not{2} \: tan \theta \: \cancel{(tan \theta + sec \theta)}}{ \not{2} \: sec \theta \: \cancel{(sec \theta + tan \theta)}} }$

$\implies \sf{ \dfrac{tan \theta}{sec \theta} }$

$\implies \sf{ \dfrac{ \sin \theta }{ \cos \theta} \times \cos \theta }$

$= \boxed{ \bf{sin \theta}}$

★ HENCE PROVED ★