Question

Prove that 1 + cot theta minus cosec theta)(1+tan theta +sec theta)=2

Answer 1

(1 + cot⁡θ - cosec⁡θ)(1 + tan⁡θ + sec⁡θ) =  2, it is proved

Step-by-step explanation:

L.H.S. = $(1 +\cot\theta + Cosec  \theta)(1+\theta +\sec \theta)\\\\=(1 + \dfrac{\cos\theta}{\sin \theta})+ (-\dfrac{1}{\sin\theta})+(\dfrac{1}{\cos\theta})\\=\dfrac{(\sin\theta+\cos\theta -1)}{\sin\theta}.\dfrac{(\cos\theta+\sin\theta+1)}{\cos\theta}\\= \dfrac{\sin\theta\times\cos\theta + \sin^2\theta+\sin^2 \theta +\cos^2\theta + \cos\theta \times \sin \theta +\cos \theta -\cos\theta -\sin \theta -1}{\sin \theta \times \cos \theta}$$=\dfrac{1+2\sin\theta\times\cos\theta}{\sin\theta\times \cos\theta}\\=\dfrac{1}{\sin\theta \times \cos\theta}+2\\=2 = \textrm{Left hand Side}$