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.

It is known that 0 < θ < 90 and it is also given that $\cos θ = \frac{24}{25}$, then find the value of $\tan(\frac{θ}{2})$?

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.