NDA Mathematics Trigonometric Functions Medium Questions

If $\cos \theta =\frac{1}{3}$, then what is the value of $\sin \left(\frac{\theta }{2}\right)\sin \left(\frac{3\theta }{2}\right)$ ?

If $x=\sec \theta -\tan \theta$ and $y=\mathrm{cosec}\theta +\cot \theta$, then which one of the following is correct ?

A plane is observed to be approaching the airport. It is at a distance of 10 km from the point of observation and makes an angle of elevation of $67.5^{\circ }$. What is the height of the plane above the ground ?

If $2\sec 4\beta =\tan 2\alpha +\cot 2\alpha$, then which one of the following is a possible value of $(\alpha +\beta )$ ?

If $\alpha$ and $\beta$ are complementary angles such that $\alpha -\beta =\frac{\pi }{6}$ and $m\tan \beta =n\tan \alpha$, then what is $\frac{m+n}{m-n}$ equal to ?

What is the length of the chord of a unit circle which subtends at the centre of the circle an angle of $45^{\circ }$ ?

The expression $\cos x+\sqrt{3}\sin x$ is maximum when $x$ is equal to

If $\theta$ lies in the fourth quadrant and $3\cot \theta +4=0$, then what is the value of $\sin 2\theta +\cos 2\theta$ ?

If $\cos \alpha +\cos \beta =0=\sin \alpha +\sin \beta$ and $\alpha$ not equals to $\beta$, then what is the value of

$\cos 2\alpha +\cos 2\beta +2\cos (\alpha +\beta )?$

What is the smallest positive x satisfying ${\log }_{\sin x}\cos x+{\log }_{\cos x}\sin x=2$ ?

If $A$ and $B$ are acute angles such that $2A+2B=\pi$, then what is the maximum value of $\sin A\sin B$ ?