Trigonometric Identities and Applications

In the identity $\sin^2 \theta + \cos^2 \theta = k$, the value of $k$ is always:

What is the value of $\frac{1 - \cos^2 A}{\sin A}$?

Find the value of $x$ if $\sin^2 25^\circ + \cos^2 25^\circ = x$.

Simplify $\cos^2 A + \frac{1}{1 + \cot^2 A}$.

In the given figure, a unit circle is centered at the origin $O$. A point $P(x, y)$ lies on the circle such that the line segment $OP$ makes an angle $\theta$ with the positive x-axis. Using the relationship between the coordinates of $P$ and the trigonometric identity $\sin^2\theta + \cos^2\theta = 1$, find the value of the expression $(\sin \theta + \cos \theta)^2 - 2\sin \theta \cos \theta$.

If $\cos \theta + \cos^2 \theta = 1$, find the value of $\sin^2 \theta + \sin^4 \theta$.

Simplify: $(\sec A + \tan A)(1 - \sin A)$.

A light beam hits a surface at an angle $A$. The reflected intensity is proportional to $I = 5 \cos^2 A$ and the absorbed intensity is proportional to $J = 5 - 5 \cos^2 A$. Find the expression for $J$ in terms of $\sin A$.

A surveyor is measuring two paths from a point $O$. Path 1 is $d_1 = 10 \sin A$ and Path 2 is $d_2 = 10 \cos A$. What is the square of the distance from the end of Path 1 to the end of Path 2 if they are perpendicular?

If $\sin \theta + \sin^2 \theta = 1$, then find the value of $\cos^2 \theta + \cos^4 \theta$.

What is the value of $\frac{1 - \tan^2 45^\circ}{1 + \tan^2 45^\circ}$?

In the given sector of a unit circle, the coordinates of $P$ are $(\cos \theta, \sin \theta)$. What is the length of the vertical line segment $PQ$?

Find the value of $(\sin 45^\circ + \cos 45^\circ)^2$.

What is $\csc^2 \theta - \cot^2 \theta$?

In the given right-angled triangle $ABC$, the angle at $B$ is $90^\circ$ and $\angle BAC = \theta$. Which of the following expressions is equivalent to $\sec^2 \theta - \tan^2 \theta$ using the sides of the triangle?

In triangle $PQR$ right angled at $Q$, if $PQ = 5$ cm and $PR = 13$ cm, find $\tan^2 P + 1$.

If $x = a \cos \theta$ and $y = b \sin \theta$, then find the value of $\frac{x^2}{a^2} + \frac{y^2}{b^2}$.

A ladder is leaning against a wall at an angle $\theta$ with the ground. The ratio of the distance from the wall to the length of the ladder is $k = \sqrt{1 - \sin^2 \theta}$. If the ladder is $10$ meters long, and $\theta = 60^\circ$, find the distance of the foot of the ladder from the wall.

A satellite tracking station determines the elevation angle $\theta$. The signal strength $S$ is proportional to $(\sec^2 \theta - 1)(\cot^2 \theta)$. What is the simplified signal strength factor?

In a signal processing task, the amplitude $A$ is defined by $A = \frac{1 + \cot^2 x}{1 + \tan^2 x}$. Which of the following is the correct simplified form of $A$?

From a point on the ground, $20\text{ m}$ away from the foot of a vertical tower, the angle of elevation of the top of the tower is $60^\circ$. The height of the tower is:

If the altitude of the sun is $60^\circ$, the height of a vertical tower that will cast a shadow of length $30\text{ m}$ is:

The angle of elevation of the top of a tower from a point on the ground, which is $30\text{ m}$ away from the foot of the tower, is $30^\circ$. The height of the tower is:

A pole $6\text{ m}$ high casts a shadow $2\sqrt{3}\text{ m}$ long on the ground, then the Sun's elevation is:

An observer stands at point $P$ on horizontal ground, $30\text{ m}$ away from the base of a vertical tower $AB$. The observer finds the angle of elevation to the top of the tower $A$ to be $60^\circ$. What is the height of the tower $AB$?

The angle of elevation of the top of a chimney from the foot of a tower is $60^\circ$ and the angle of elevation of the top of the tower from the foot of the chimney is $30^\circ$. If the tower is $50\text{ m}$ high, find the height of the chimney.

From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are $30^\circ$ and $45^\circ$ respectively. If the bridge is at a height of $3\text{ m}$ from the banks, find the width of the river.

The angle of elevation of the top of a tower from two points at a distance of 4 m and 9 m from the base of the tower and in the same straight line with it are complementary. Find the height of the tower.

As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are $30^{\circ}$ and $45^{\circ}$. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

A TV tower stands vertically on a bank of a canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is $60^{\circ}$. From another point 20 m away from this point on the line joining this point to the foot of the tower, the angle of elevation of the top of the tower is $30^{\circ}$. Find the width of the canal.