Question

Prove : 1+ cot2 Ө/1coseӨo+= CosecӨ​

Answer 1

Answer in Attachment..

Answer 2

Question ; Prove :

$\longmapsto\sf 1 + \dfrac{cot^2 \theta}{1 + cosec \theta} = cosec \theta$

Identities used :

$\bullet\sf \ \ \ \dfrac{1}{sin \theta} = cosec \theta \\\\\\ \bullet\sf \ \ \ cot^2 \theta = \dfrac{cos^2 \theta}{sin^2 \theta}$

Proof :

$\longmapsto\sf 1 + \dfrac{cot^2 \theta}{1 + cosec \theta} \\\\\\ \longmapsto\sf \dfrac{1 + cosec \theta + cot^2 \theta}{1 + cosec \theta} \\\\\\ \longmapsto\sf \dfrac{ 1 + \dfrac{1}{sin \theta} + \dfrac{cos^2 \theta}{sin^2 \theta}}{1 + \dfrac{1}{sin \theta}} \\\\\\ \longmapsto\sf \dfrac{\dfrac{sin^2 \theta + sin \theta + cos^2 \theta}{sin^2 \theta}}{ \dfrac{sin \theta + 1}{sin \theta}} \\\\\\ \longmapsto\sf \dfrac{ \dfrac{1 + sin \theta}{sin^2 \theta}}{ \dfrac{sin \theta + 1}{sin \theta}}$

$\bullet\underline{\sf{\ \ Identity \: : \ sin^2 \theta + cos^2 \theta = 1 \ }}$

$\longmapsto\sf \dfrac{ \cancel{1 + sin \theta} }{\cancel{sin^2 \theta}} \ \times \ \dfrac{ \cancel{sin \theta} }{\cancel{1 + sin \theta}} \\\\\\ \longmapsto\sf \dfrac{1}{sin \theta} \\\\\\ \bullet\underline{\sf{ \ Identity \ : \ \dfrac{1}{sin \theta} = cosec \theta}} \\\\\\ \longmapsto\large\underline{\boxed{\sf{ cosec \theta }}} \\\\\\ \longmapsto\sf RHS \\\\\\ \therefore\underline{\sf{LHS \ = RHS \ \ \ [Proved] }}$