Question

If a straight line makes angles $\alpha , \beta , \gamma $ with the coordinate axes, then $\frac{1-\tan^{2} \alpha }{1+tan^{2} \alpha} +\frac{1}{sec 2 \beta} -2\sin^{2} \gamma =$

• A. -1
• B. 1
• C. -2
• D. 2

Answer 1

The Correct Option is C

If a straight line makes angles $\alpha, \beta, \gamma$ with the coordinate axes.
Then, $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$
$\Rightarrow \frac{1+\cos 2 \alpha}{2}+\frac{1+\cos 2 \beta}{2}+1-\sin ^{2} \gamma=1$
$\Rightarrow 1+\cos 2 \alpha+1+\cos 2 \beta+2-2 \sin ^{2} \gamma=2$
$\Rightarrow \cos 2 \alpha+\cos 2 \beta-2 \sin ^{2} \gamma=-2$
$\Rightarrow \frac{1-\tan ^{2} \alpha}{1+\tan ^{2} \alpha}+\frac{1}{ sec\, 2 \beta}-2 \sin ^{2} \gamma=-2$