List of top Quantitative Aptitude Questions on Trigonometric Identities

In $ \triangle ABC $, $ \angle B = 90^\circ $, $ BC = 6 \, \text{cm} $, and $ AC = 10 \, \text{cm} $. What is the value of $ (\sin 2A - \cos 2A) $?

\[ \frac{\tan^2 \theta + \cot^2 \theta + 2}{\sec \theta \csc \theta}, \ 0^\circ<\theta<90^\circ, \text{ is equal to:} \]

Two persons, on either side of a tower 45 m high, observe the angles of elevation of the top of the tower to be 30° and 60°, respectively. The distance (in m) between the two persons is:

If
$ \sec(5^\circ - 2\theta) = \csc(5\theta - 5^\circ),\ 0^\circ < \theta < 90^\circ $,
then what will be the value of
$ \frac{\sin 3\theta + \tan 2\theta}{\sec 2\theta} $?

The expression $ \frac{(2\sin^2\theta - 1)(1 + \tan^2\theta)(\sec\theta + \tan\theta)(1 - \sin\theta)}{(\sec^2\theta - 1)(\tan\theta - \cot\theta)\cos^2\theta} $ for $ 0^\circ<\theta<90^\circ $ is equal to:

Evaluate: \[ \frac{\sin \theta (1 + \tan \theta) + \cos \theta (1 + \cot \theta)}{(\cos \theta - \sin \theta)(\sec \theta - \cos \theta)(\tan \theta + \cot \theta)} \]

From a point on a bridge across a river, the angles of depressions of the banks on opposite sides of the river are $30^\circ$ and $60^\circ$, respectively. If the height of the bridge from the banks is $ h $ metres and the width of the river is $ k $ metres, then $ h : k $ is equal to:

The expression $ \frac{\sec^6 \theta - \tan^6 \theta - 3 \sec^2 \theta \tan^2 \theta}{1 + 2 \sin^2 \theta - \sin^4 \theta + \cos^4 \theta} $ is equal to:

If $ 3\sin^2 x + 10\cos x - 6 = 0 $, $ 0^\circ<x<90^\circ $, then the value of $ \sec x + \cosec x + \cot x $ is:

X and Y are two points that are 135 m apart on the ground on either side of a pole and in the same line. The angles of elevation of a bird sitting on the top of the pole from X and Y are $ 30^\circ $ and $ 60^\circ $ respectively. The distance of Y from the foot of the pole (in m) is:

If $ \sec \theta + \tan \theta = p $, then $ \frac{\sin \theta - 1}{\sin \theta + 1} $ is equal to:

If $ \sec \theta = a + \frac{1}{4a^2} $, $ 0^\circ<\theta<90^\circ $, then $ \csc \theta + \cot \theta = $:

The value of $ \frac{(1 + \cot \theta - \csc \theta)(1 + \tan \theta + \sec \theta)}{\tan^2 \theta + \cot^2 \theta - \sec^2 \theta \csc^2 \theta} $ is:

The expression $ \frac{(1 + \tan \theta) \cos \theta}{\sin \theta \tan \theta (1 - \tan \theta) + \sin \theta \sec^2 \theta} $ is equal to:

The value of $ \frac{\sin^2 \theta (2 + \cot^2 \theta) - \sin^2 \theta + 2}{\tan^2 \theta + \cot^2 \theta - \sec^2 \theta \csc^2 \theta} $ is:

If $ \sec \theta + \tan \theta = p $, then $ \frac{\sin \theta - 1}{\sin \theta + 1} $ is equal to:

The value of $ \frac{\sin^2 \theta (2 + \cot^2 \theta) - \sin^2 \theta + 2}{\tan^2 \theta + \cot^2 \theta - \sec^2 \theta \csc^2 \theta} $ is:

From a point on a bridge across a river, the angles of depressions of the banks on opposite sides of the river are $30^\circ$ and $60^\circ$, respectively. If the height of the bridge from the banks is $ h $ metres and the width of the river is $ k $ metres, then $ h : k $ is equal to:

The expression $ \frac{(1 + \tan \theta) \cos \theta}{\sin \theta \tan \theta (1 - \tan \theta) + \sin \theta \sec^2 \theta} $ is equal to:

The expression $ \frac{\sec^6 \theta - \tan^6 \theta - 3 \sec^2 \theta \tan^2 \theta}{1 + 2 \sin^2 \theta - \sin^4 \theta + \cos^4 \theta} $ is equal to:

If $ 3 \sin^2 x + 10 \cos x - 6 = 0, 0^\circ<\theta<90^\circ $, then the value of $ \sec x + \csc x + \cot x $ is:

If $ \sec \theta = a + \frac{1}{4a}, 0^\circ<\theta<90^\circ $, then $ \csc \theta + \cot \theta = $:

Simplify the expression: \[ \frac{\sin \theta(1 + \tan \theta) + \cos \theta (1 + \cot \theta)}{\csc \theta - \sin \theta} \cdot \frac{\sec \theta}{\cos \theta (\tan \theta + \cot \theta)} = ? \]

The value of $ \frac{(1 + \cot \theta - \csc \theta)(1 + \tan \theta + \sec \theta)}{\tan^2 \theta + \cot^2 \theta - \sec^2 \theta \csc^2 \theta} $ is:

X and Y are the two points that are 135 m apart on the ground on either side of a pole and in the same line. The angles of elevation of a bird sitting on the top of the pole from X and Y are 30° and 60° respectively. The distance of Y from the foot of the pole (in m) is: