Question

(cos theta cosec theta - sin theta sec theta) / cos theta + sin theta = cosec theat - sec theat ​

Answer 1

Prove that :-

$\sf \: \dfrac{cos \theta.cosec \theta- sin\theta.sec\theta}{cos\theta + sin\theta} = cosec\theta - sec\theta$

some informations :-

$\sf \blue \bigstar \: \: cosec \theta \: = \dfrac{1}{sin\theta} \\ \\ \sf \blue \bigstar \: \:sec \theta\: = \frac{1}{cos\theta}$

$\sf \blue \bigstar$ cos²θ − sin²θ = (cosθ + sinθ)(cosθ − sinθ)

proof :-

LHS

$\sf \: \dfrac{cos \theta.cosec \theta- sin\theta.sec\theta}{cos\theta + sin\theta} \\ \\ \sf \blue \implies \dfrac{cos \theta. \dfrac{1}{sin \theta} - sin \theta. \dfrac{1}{cos \theta} }{cos \theta + sin \theta} \\ \\ \sf \blue \implies \: \dfrac{ \dfrac{cos \theta}{sin\theta} - \dfrac{sin\theta}{cos\theta} }{cos\theta + sin\theta} \\ \\ \bf \: taking \: LCM \\ \\ \sf \blue \implies \: \dfrac{ \dfrac{ {cos \theta}^{2} - {sin}^{2} \theta}{sin \theta.cos \theta} }{cos \theta + sin \theta}$

using :- a² – b² = (a + b)(a – b)

$\sf \blue \implies \: \dfrac{ \dfrac{(cos\theta + sin\theta)(cos\theta - sin\theta)}{sin\theta.cos\theta} }{cos\theta + sin\theta} \\ \\ \sf \blue \implies \: \dfrac{(cos\theta + sin\theta)(cos\theta - sin\theta)}{sin\theta.cos\theta} \times \dfrac{1}{cos\theta + sin\theta} \\ \\ \sf \blue \implies \: \dfrac{(\cancel{cos\theta + sin\theta})(cos\theta - sin\theta)}{cos\theta.sin\theta} \times \dfrac{1}{ \cancel{cos\theta + sin\theta} } \\ \\ \sf \blue \implies \: \frac{cos\theta - sin\theta}{cos\theta.sin\theta} \\ \\ \sf \blue \implies \: \frac{cos \theta}{cos\theta.sin\theta} - \frac{sin \theta}{cos\theta.sin\theta} \\ \\ \sf \blue \implies \: \frac{ \cancel{cos \theta}}{\cancel{cos\theta}.sin\theta} - \frac{\cancel{sin \theta}}{cos\theta.\cancel{sin\theta} } \\ \\ \sf \blue \implies \: \frac{1}{sin \theta} - \frac{1}{cos \theta} \\ \\ \sf \blue \implies \: cosec \theta \: - sec \theta$

= RHS

hence,

LHS = RHS

some more formula :-

★ sinθ/cosθ = tanθ

★ cosθ/sinθ = cotθ

★ sin²θ + cos²θ = 1

★ 1 – sin²θ = cos²θ

★ 1 – cos²θ = sin²θ

★ 1 + tan²θ = sec²θ

★ sec²θ – 1 = tan²θ

★ sec²θ – tan²θ = 1

★ 1 + cot²θ = cosec²θ

★ cosec²θ – 1 = cot²θ

★ cosec²θ – cot²θ = 1