Prove that : 1 + sec θ – tan θ / 1 + sec θ + tan θ = 1 – sin θ / cos θ

Question

Q)  Prove that : $\frac{1 + \sec 	heta - 	an 	heta} {1 + \sec 	heta + 	an 	heta} = \frac{1 - \sin 	heta} {\cos 	heta}$

Sapling Academy · Accepted Answer

Let's start from simplifying the LHS:

LHS = $\frac{(1 + \sec 	heta - 	an 	heta)} {(1 + \sec 	heta + 	an 	heta)}$

= $\frac{(1 + \frac{1}{\cos 	heta} - \frac{\sin 	heta)}{\cos 	heta} )} {(1 + \frac{1}{\cos 	heta} + \frac{\sin 	heta)}{\cos 	heta} )}$

= $\frac{\cos 	heta + 1 - \sin 	heta} {\cos 	heta + 1 + \sin 	heta}$

Since, we need to get demnominator in simplified form, hence let's multiply nominator and denominator by cos θ + sin θ - 1, we get:

LHS = $\frac{\cos 	heta - \sin 	heta + 1} {\cos 	heta + \sin 	heta + 1} 	imes \frac{\cos 	heta + \sin 	heta - 1}{\cos 	heta + \sin 	heta -1}$

= $\frac{(\cos 	heta + 1 - \sin 	heta)	imes (\cos 	heta + \sin 	heta - 1 )}{(\cos 	heta + 1 + \sin 	heta) 	imes ((\cos 	heta + \sin 	heta - 1)}$

= $\frac{(\cos 	heta + (1 - \sin 	heta))	imes (\cos 	heta - (1 - \sin 	heta))}{((\cos 	heta + \sin 	heta) + 1) 	imes ((\cos 	heta + \sin 	heta) - 1)}$

= $\frac{(\cos^2 	heta - (1 - \sin 	heta)^2)}{((\cos 	heta + \sin 	heta)^2 - 1)}$

= $\frac{(\cos^2 	heta - (1 + \sin^2 	heta - 2 \sin 	heta))}{((\cos^2 	heta + \sin^2 	heta + 2 \sin 	heta \cos 	heta) - 1)}$

= $\frac{(\cos^2 	heta - 1 - \sin^2 	heta + 2 \sin 	heta)}{((\cos^2 	heta + \sin^2 	heta + 2 \sin 	heta \cos 	heta) - 1)}$

We know that sin2 θ + cos2 θ = 1

∴ LHS = $\frac{(\cos^2 	heta - (\cos^2 	heta + \sin^2 	heta) - \sin^2 	heta + 2 \sin 	heta)}{((1 + 2 \sin 	heta \cos 	heta) - 1)}$

= $\frac{(\cos^2 	heta - \cos^2 	heta - \sin^2 	heta - \sin^2 	heta + 2 \sin 	heta)}{(1 + 2 \sin 	heta \cos 	heta - 1)}$

= $\frac{(\cancel{\cos^2 	heta} - \cancel{\cos^2 	heta} - \sin^2 	heta - \sin^2 	heta + 2 \sin 	heta)}{(\cancel{1} + 2 \sin 	heta \cos 	heta - \cancel{1})}$

= $\frac{(- 2 \sin^2 	heta + 2 \sin 	heta)}{(2 \sin 	heta \cos 	heta )}$

= $\frac{2 \sin 	heta (1 - \sin 	heta)}{(2 \sin 	heta \cos 	heta )}$

= $\frac{(1 - \sin 	heta)}{(\cos 	heta )}$

Hence Proved !

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