Ratios of the Difference of Two Angles

Simplify the following trigonometric expression as much as possible: $\cos(x) \cos(y) + \sin(x) \sin(y)$. Then, use the simplified result to evaluate the expression exactly when $x = 75^\circ$ and $y = 15^\circ$.

Cian is solving an exercise where he has to calculate the exact value of $\cos(15^\circ)$ by applying the cosine difference formula with $15^\circ = 60^\circ - 45^\circ$. On his exam, he writes the following working:

When checking his result with a calculator, he is surprised to find that it gives a negative number (approximately $-0.26$), which he knows is impossible because $15^\circ$ lies in the first quadrant and its cosine must be positive.

1. a) Identify precisely the mistake Cian made and explain, based on the parity of trigonometric functions ($\cos(-x) = \cos x$ and $\sin(-x) = -\sin x$), why a central positive sign appears in the cosine difference formula.
2. b) Correct the calculation and find the exact, simplified value of $\cos(15^\circ)$.

Prove analytically that $\tan(45^\circ - x) = \frac{1 - \tan x}{1 + \tan x}$. Use this result to find the exact value of $\tan(15^\circ)$ given that $\tan(30^\circ) = \frac{\sqrt{3}}{3}$.

Simplify the expression $\sin(90^\circ - x)$ by applying the formula for the sine of the difference of two angles and verify that it is equivalent to one of the basic trigonometric ratios of the angle $x$.

Prove step by step that $\cos(180^\circ - \alpha) = -\cos \alpha$ using the formula for the cosine of the difference of two angles.

Simplify the following algebraic trigonometric expression and express it purely in terms of the tangents of the angles $a$ and $b$:

$\frac{\sin(a - b)}{\cos a \cos b}$

Given that two angles $\alpha$ and $\beta$ are complementary ($\alpha + \beta = 90^\circ$), prove step by step that the identity $\sin(\alpha - \beta) = \cos(2\beta)$ holds.

Calculate the exact value of $\cos(15^\circ)$ without using a calculator, applying the formula for the cosine of the difference of two angles and knowing that $15^\circ = 45^\circ - 30^\circ$.

Find the exact value of $\sin(15^\circ)$ by expressing the angle as the difference $60^\circ - 45^\circ$ and applying the corresponding identity.

Determine the exact value of $\tan(15^\circ)$ from the tangents of the special angles $45^\circ$ and $30^\circ$. Rationalise the result if necessary.

Calculate the exact value of $\tan(105^\circ)$ using the formula for the difference of two angles, considering that $105^\circ = 135^\circ - 30^\circ$.

Evaluate the following trigonometric expression exactly, without using a calculator, by identifying which expansion it corresponds to:

$\sin(75^\circ) \cos(15^\circ) - \cos(75^\circ) \sin(15^\circ)$

Given that $\sin \alpha = \frac{4}{5}$ and $\cos \beta = \frac{12}{13}$, and that both $\alpha$ and $\beta$ are angles in the first quadrant, calculate the exact value of $\sin(\alpha - \beta)$.

In a maritime navigation simulation, a ship must adjust its current heading. The necessary correction angle is given by the expression $\alpha - \beta$. Given that $\alpha$ and $\beta$ are acute angles with $\tan \alpha = 3$ and $\tan \beta = \frac{1}{2}$, calculate the exact value of $\tan(\alpha - \beta)$ and deduce the size of the correction angle.

Solve the following trigonometric equation, finding all possible solutions in the interval $0^\circ \leq x < 360^\circ$:

$\sin(x - 30^\circ) = \frac{\sqrt{3}}{2}$

Solve the following trigonometric equation, determining all possible solutions in the interval $[0, 2\pi)$ radians:

$\cos(x - 45^\circ) + \cos(x + 45^\circ) = 1$

A light ray passes through two different media with angles of refraction $\alpha$ and $\beta$. If, from the material properties, we know that $\cos \alpha = \frac{3}{5}$ (where $\alpha$ is an angle in the fourth quadrant) and $\sin \beta = \frac{5}{13}$ (where $\beta$ is an angle in the second quadrant), determine the exact value of $\cos(\alpha - \beta)$.

In a market study, the seasonal sales cycles of two products are modelled using trigonometric functions. The phase difference that determines direct competition is given by the value of $\cos(x - y)$. If $\sin x = \frac{1}{3}$ and $\cos y = \frac{2}{5}$, with $x$ located in the first quadrant and $y$ in the fourth quadrant, find the exact value of $\cos(x - y)$.

A surveyor observes a cliff of height $H$ and, located right on the edge above it, a lighthouse of height $h$. From their position at a horizontal distance $d$, the angle of elevation to the base of the lighthouse is $\beta$ and to the top of the lighthouse is $\alpha$. Express the tangent of the angle subtended by the lighthouse alone, that is $\tan(\alpha - \beta)$, purely in terms of the distances $H$, $h$ and $d$.

In an alternating current circuit, the potential difference between two nodes is given by the superposition of two out-of-phase signals: $V(t) = 10 \sin(\omega t) - 10 \sin(\omega t - \frac{\pi}{3})$. Use the trigonometric difference angle formulas to express the potential difference $V(t)$ in the simplified form $A \sin(\omega t + \phi)$, determining the values of $A$ and $\phi$.