Question

The simplified value of $(\frac{1-\sin \alpha }{\cos \alpha }+\frac{\cos \alpha }{1+\sin \alpha })(\sec \alpha +\frac{1}{\cot \alpha })$ is _____

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is C $2$

$(\frac{1-\sin \alpha }{\cos \alpha }+\frac{\cos \alpha }{1+\sin \alpha })(\sec \alpha +\frac{1}{\cot \alpha })=\left(\frac{(1-\sin \alpha )(1+\sin \alpha )+{\cos }^2\alpha }{(1+\sin \alpha )\cos \alpha }\right)(\frac{1}{\cos \alpha }+\frac{\sin \alpha }{\cos \alpha })$
$=\left(\frac{(1-{\sin }^2\alpha +{\cos }^2\alpha }{(1+\sin \alpha )\cos \alpha }\right)\left(\frac{1+\sin \alpha }{\cos \alpha }\right)=\left(\frac{{\cos }^2\alpha +{\cos }^2\alpha }{\cos \alpha }\right)\left(\frac{1}{\cos \alpha }\right)=2$