4.    Choose the correct option. Justify your choice.

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

       (A) 0    (B) 1    (C) 2    (D) –1

Answers (1)
M manish

The correct option is (C)

(1+\tan \theta +\sec \theta )(1+\cot \theta -cosec\: \theta ).......................(i)

we can write his above equation as;
\\=(1+\sin \theta/\cos \theta +1/\cos \theta )(1+\cos\theta/\sin \theta -1/sin\theta )\\\\= \frac{(1+\sin\theta+\cos\theta)}{\cos\theta.\sin\theta}\times (\frac{(\sin\theta+\cos\theta-1}{\sin\theta.\cos\theta})\\\\= \frac{(\sin\theta+\cos\theta)^2-1^2}{\sin\theta.\cos\theta}\\\\= \frac{\sin^2\theta+\cos^2\theta+2\sin\theta.\\cos\theta-1}{\sin\theta.\cos\theta}\\\\= 2\times\frac{\sin\theta.\cos\theta}{\sin\theta.\cos\theta}
= 2

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