Question

$(1+tan\theta +sec\theta )(1+cot\theta -cosec\theta )=$

• A. 0
• B. 1
• C. 2
• D. 12

Answer 1

Answer: (C)

2

$(1+\tan \theta +\sec \theta )(1+\cot \theta -\csc \theta )$

$=1(1+\cot \theta -\csc \theta )+\tan \theta (1+\cot \theta -\csc \theta )+\sec \theta (1+\cot \theta -\csc \theta )$

$=1+\cot \theta -\csc \theta +\tan \theta +\tan \theta \cot \theta -\tan \theta \csc \theta +\sec \theta +\sec \theta \cot \theta -\sec \theta \csc \theta$

$=1+\cot \theta -\csc \theta +\tan \theta +1-\frac{\sin \theta }{\cos \theta }\times \csc \theta +sec\theta +\frac{1}{\cos \theta }\times \frac{\cos \theta }{\sin \theta }-\sec \theta \csc \theta$

$=1+\cot \theta -\csc \theta +\tan \theta +1-\frac{\sin \theta }{\cos \theta }\times \frac{1}{\sin \theta }+sec\theta +\frac{1}{\cos \theta }\times \frac{\cos \theta }{\sin \theta }-\sec \theta \csc \theta$

$=1+\cot \theta -\csc \theta +\tan \theta +1-\sec \theta +sec\theta +\csc \theta -\sec \theta \csc \theta$

$=2+\cot \theta +\tan \theta -\sec \theta \csc \theta$

$=2+\frac{\cos \theta }{\sin \theta }+\frac{\sin \theta }{\cos \theta }-\sec \theta \csc \theta$

$=2+\frac{{\cos }^2\theta +{\sin }^2\theta }{\sin \theta \cos \theta }-\sec \theta \csc \theta$

$=2+\frac{1}{\sin \theta \cos \theta }-\sec \theta \csc \theta$

$=2+\sec \theta \csc \theta -\sec \theta \csc \theta$

$=2$