Sum Formulas for Sine, Cosine, and Tangent

Express $\cos(x + \frac{\pi}{2})$ in terms of the trigonometric ratios of the angle $x$ by expanding the corresponding sum formula and simplifying the result.

Calculate the exact value of $\sin(75^\circ)$ by decomposing the angle as the sum of two notable angles ($45^\circ$ and $30^\circ$).

Determine the exact value of $\cos(105^\circ)$ using the formula for the cosine of a sum, knowing that $105^\circ = 60^\circ + 45^\circ$.

Find the exact value of $\tan(75^\circ)$ by applying the formula for the tangent of a sum from the angles of $45^\circ$ and $30^\circ$. Rationalize the denominator if necessary.

Simplify the following trigonometric expression to a single exact numerical term by applying the addition formulas in reverse:

$\sin(22^\circ)\cos(8^\circ) + \cos(22^\circ)\sin(8^\circ)$

Reduce the following expression to an exact numerical value:

$\cos(40^\circ)\cos(20^\circ) - \sin(40^\circ)\sin(20^\circ)$

Calculate exactly the value of the expression $\sin(15^\circ)\cos(30^\circ) + \cos(15^\circ)\sin(30^\circ)$. Use this geometric result to find the exact area of a parallelogram whose adjacent sides measure $8$ in and $10$ in, knowing that the angle they form is precisely the sum of $15^\circ$ and $30^\circ$.

Knowing that $\sin \alpha = \frac{4}{5}$ and $\cos \beta = \frac{12}{13}$, and that both $\alpha$ and $\beta$ are acute angles (belonging to the first quadrant), calculate the exact value of $\sin(\alpha + \beta)$.

Given the angles $\alpha$ and $\beta$ such that $\sin \alpha = \frac{3}{5}$ with $\alpha \in [\frac{\pi}{2}, \pi]$ and $\cos \beta = \frac{5}{13}$ with $\beta \in [\frac{3\pi}{2}, 2\pi]$, calculate the exact value of $\cos(\alpha + \beta)$.

A surveyor observes the roof of a building with an angle of elevation $\alpha$, satisfying $\cos \alpha = \frac{4}{5}$. Behind this building, in the same line of sight, there is a telecommunications tower. The additional angle of elevation from the roof of the building to the top of the tower, measured from the previous line of sight, is $\beta$, with $\sin \beta = \frac{7}{25}$. Knowing that both angles are acute, calculate the exact value of the cosine of the total angle of elevation $\alpha + \beta$ to the top of the tower.

Prove the following trigonometric identity by expanding the numerator using the corresponding formula and simplifying the fraction:

$\frac{\sin(\alpha+\beta)}{\cos\alpha \cos\beta} = \tan\alpha + \tan\beta$

Simplify the trigonometric expression $\cos(x + \frac{\pi}{3}) + \sin(x + \frac{\pi}{6})$ so that it is exclusively in terms of $\cos x$ and/or $\sin x$.

In any non-right triangle $ABC$, knowing that the sum of its interior angles is $A+B+C = 180^\circ$, prove mathematically the following fundamental relationship:

$\tan A + \tan B + \tan C = \tan A \tan B \tan C$

Given two angles $\alpha$ and $\beta$ such that $\tan \alpha = 3$ and $\tan \beta = 2$, determine the exact value of $\tan(\alpha + \beta)$.

In a solar engineering project, a photovoltaic panel has an initial inclination $\alpha$ such that $\tan \alpha = \frac{3}{4}$. To optimize capture in winter, a mechanism increases the inclination by an additional angle of $15^\circ$. Calculate the exact value of $\tan(\alpha + 15^\circ)$ without using decimal approximations, knowing that $\tan(15^\circ) = 2 - \sqrt{3}$.

In an optics experiment, a light beam passes through two consecutive prisms that deviate its path in the same plane. The first prism produces a deviation $\alpha$ such that $\tan \alpha = \frac{1}{3}$, and the second an additional deviation $\beta$ such that $\tan \beta = \frac{1}{2}$. Show that the total deviation $\alpha + \beta$ is exactly $45^\circ$ by calculating the tangent of the sum angle.

Knowing that $\tan \alpha = \frac{1}{2}$ and that the tangent of the sum angle is $\tan(\alpha + \beta) = 3$, determine the exact value of $\tan \beta$ by setting up the corresponding equation from the addition formula.

A planar robotic arm consists of two links. The first link forms an angle $\alpha$ with the horizontal, where $\sin \alpha = \frac{\sqrt{2}}{2}$ (being in the first quadrant). The second link forms an angle $\beta$ of relative rotation with respect to the extension of the first link. If to reach a critical part it is required that the total inclination of the second link with respect to the horizontal is such that $\tan(\alpha + \beta) = 2$, calculate the exact value of $\tan \beta$.

A reconnaissance plane initially flies on a heading that forms an angle $\alpha$ north of east, verifying that $\tan \alpha = \frac{3}{4}$. A strong crosswind deviates its path by an additional angle $\beta$ to the north, so that the effective heading over the ground forms an angle $\alpha + \beta$ with the east. If the control tower measures that the tangent of the effective heading is $\frac{12}{5}$, find the exact values of $\sin \beta$ and $\cos \beta$ without resorting to calculating the angles with a calculator.

Solve the trigonometric equation $\sin(x + 30^\circ) = 2\cos x$ for the interval $x \in [0^\circ, 360^\circ)$. Express the solutions exactly in degrees.

Find all exact solutions of the equation $\sin(x + \frac{\pi}{6}) - \cos x = 0$ in the interval $[0, 2\pi)$.

Solve the following system of trigonometric equations for $x, y \in [0, \frac{\pi}{2}]$ and give the exact values of $x$ and $y$ in radians:

$\sin(x+y) = 1$

$\cos x \sin y = \frac{1}{2}$