The value of $(1 + an heta + \sec heta)(1 + \cot heta - \csc heta)$ is:

Q: The value of $(1 + \tan \theta + \sec \theta)(1 + \cot \theta - \csc \theta)$ is:

Convert all terms to sine and cosine: $(1 + \frac{\sin \theta}{\cos \theta} + \frac{1}{\cos \theta})(1 + \frac{\cos \theta}{\sin \theta} - \frac{1}{\sin \theta}) = (\frac{\cos \theta + \sin \theta + 1}{\cos \theta})(\frac{\sin \theta + \cos \theta - 1}{\sin \theta})$. Let $x = \cos \theta + \sin \theta$, then the expression becomes $\frac{(x+1)(x-1)}{\sin \theta \cos \theta} = \frac{x^2 - 1}{\sin \theta \cos \theta} = \frac{(\cos \theta + \sin \theta)^2 - 1}{\sin \theta \cos \theta} = \frac{\cos^2 \theta + \sin^2 \theta + 2\sin \theta \cos \theta - 1}{\sin \theta \cos \theta} = \frac{1 + 2\sin \theta \cos \theta - 1}{\sin \theta \cos \theta} = \frac{2\sin \theta \cos \theta}{\sin \theta \cos \theta} = 2$.

Question

The value of $(1 + 	an 	heta + \sec 	heta)(1 + \cot 	heta - \csc 	heta)$ is:

Accepted Answer

Convert all terms to sine and cosine: $(1 + \frac{\sin 	heta}{\cos 	heta} + \frac{1}{\cos 	heta})(1 + \frac{\cos 	heta}{\sin 	heta} - \frac{1}{\sin 	heta}) = (\frac{\cos 	heta + \sin 	heta + 1}{\cos 	heta})(\frac{\sin 	heta + \cos 	heta - 1}{\sin 	heta})$. Let $x = \cos 	heta + \sin 	heta$, then the expression becomes $\frac{(x+1)(x-1)}{\sin 	heta \cos 	heta} = \frac{x^2 - 1}{\sin 	heta \cos 	heta} = \frac{(\cos 	heta + \sin 	heta)^2 - 1}{\sin 	heta \cos 	heta} = \frac{\cos^2 	heta + \sin^2 	heta + 2\sin 	heta \cos 	heta - 1}{\sin 	heta \cos 	heta} = \frac{1 + 2\sin 	heta \cos 	heta - 1}{\sin 	heta \cos 	heta} = \frac{2\sin 	heta \cos 	heta}{\sin 	heta \cos 	heta} = 2$.

NCERT · Class 10 · Mathematics · Introduction to TrigonometryThe value of $(1 + \tan \theta + \sec \theta)(1 + \cot \theta - \csc \theta)$ is: