Question

IS LHS=RHS ?

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

Say yes or no.

• A. Yes
• B. No
• C. Ambiguous
• D. Data insufficient

Answer 1

The correct option is A Yes

$LHS=\frac{1+\cos \theta -\sin \theta }{1+\sin \theta +\cos \theta }+\frac{1+\sin \theta +\cos \theta }{1+\cos \theta -\sin \theta }$
$=\frac{[1+\cos \theta -\sin \theta {]}^2+[1+\sin \theta +\cos \theta {]}^2}{(1+\cos \theta {)}^2-{\sin }^2\theta }$
$=\frac{2+2\cos \theta +2+2\cos \theta }{(1-{\sin }^2\theta )+{\cos }^2\theta +2\cos \theta }=\frac{4(1+\cos \theta )}{2{\cos }^2\theta +2\cos \theta }=\frac{4(1+\cos \theta )}{2\cos \theta (1+\cos \theta )}$
$=\frac{2}{\cos \theta }=2\sec \theta =RHS$