Trigonometry Questions for CAT

Simplify $\frac{1}{\cosec^2\theta}$.

Given $\tan(-10^\circ) \tan(-20^\circ) \tan(-30^\circ) \tan(91^\circ) \tan(92^\circ) = p,$ find the value of $\tan(87^\circ)$.

Simplify $\sin A \cos B + \cos A \sin B$.

Simplify $\sin 2\theta$ when $\sin \theta = \tfrac{3}{5}$ and $\cos \theta = \tfrac{4}{5}$.

Simplify $\sin^2 \theta + \cos^2 \theta$.

Compute $\tan \tfrac{7\pi}{6}$.

Evaluate $\log(\sin 45^\circ)+\log(\cos 45^\circ)$.

Evaluate $\cos \tfrac{\pi}{5} + \cos \tfrac{3\pi}{5} + \cos \tfrac{7\pi}{5} + \cos \tfrac{9\pi}{5}$.

The expression $12\cos A + 9\sin A − 7$ attains what maximum and minimum values?

A man walks towards a vertical building. The angle of elevation of the top of the building increases from $30^{\circ}$ to $45^{\circ}$ in 12 minutes. Find the time taken by the man to reach the building after the angle becomes $45^{\circ}$.

If $\tan x + \tan y = \tan2x$ and $x+y=45^\circ$, find $\tan x \cdot \tan y$.

Two points $A$ and $B$ are $1\,\text{km}$ apart on level ground. A drone is somewhere between them. From $A$ the elevation is $45^\circ$ and from $B$ it is $30^\circ$. The drone lies between $A$ and $B$. Find its height.

In a right-angled triangle with height ‘p’, base ‘b’, and hypotenuse ‘h’, which of the following values cannot be taken by $h^2$, assuming that p and b are positive integers?

Which of the following equals $\dfrac{1+\sin\alpha}{1-\sin\alpha}$ ?

A building is 100 m high. A drone is hovering west of the building, seeing the top at 30° elevation. Another drone is located north of the building, seeing the top at 45° elevation. Find the shortest distance between the two drones.

In $\triangle PQR$, if inradius is $r$, circumradius is $R$ and $\angle P = 90^\circ$, then find $\dfrac{PM}{QN}$ where $PM, QN$ are altitudes.

Given that cos A + cos²A = 1 and a sin¹²A + b sin¹⁰A + c sin⁸A + d sin⁶A − 1 = 0, determine the value of (a+b)/(c+d).

How many ordered pairs $(x,y)$ satisfy $\sin x - \sin y = \sin(x+y)$ and $|x|+|y|=\pi$?

On top of a building of height $8\sqrt{3}\,\text{m}$ a flag is hoisted. A man of height $3\sqrt{3}\,\text{m}$ stands $3\sqrt{10}\,\text{m}$ away from the building on level ground. He finds that the angles of elevation to the top and bottom of the flagpole are complementary. Find the height of the flagpole.

Let $x+y=\dfrac{\pi}{2}$ with $x,y\in\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]$. How many ordered pairs $(x,y)$ satisfy $\sin x + \sin y = \sin(x+y)$?