Question

1 - sin∅/1 + sin∅=(sec ∅ - tan∅)²​

Answer 1

LHS

$\to \sf \dfrac{1-sin\ \theta}{1+sin\ \theta }$

$\to \sf \dfrac{1-sin\ \theta}{1+sin\ \theta}\times \dfrac{1-sin\ \theta}{1- sin\ \theta}$

• (a + b) (a - b) = a² - b²

$\to \sf \dfrac{(1-sin\ \theta)^2}{1-sin^2\theta}$

• 1 - sin²θ = cos²θ

$\to \sf \dfrac{1+sin^2\theta-2sin \theta}{cos^2\theta}$

$\to \sf \dfrac{1}{cos^2 \theta}+\dfrac{sin^2\theta}{cos^2\theta} - \dfrac{2.sin \theta}{cos^2 \theta}$

$\to \sf sec^2\theta+tan^2\theta -2\ sec \theta .tan \theta$

$\leadsto \sf (sec\ \theta -tan\ \theta)^2\ \; \orange{\bigstar}$

RHS