📏 trigonometry

1. Problem: Simplify each expression involving trigonometric functions of angle $\alpha$. 2. a) $1 - \cos^2 \alpha - \sin^2 \alpha = 1 - (\cos^2 \alpha + \sin^2 \alpha) = 1 - 1 = 0

1. We are given the function $g(\theta) = \sqrt{3} \cos \theta - \sin \theta$ and need to show that it can be rewritten as $g(\theta) = 2 \cos \left( \theta + \frac{\pi}{6} \right)

1. **State the problem:** Solve the equation $$\tan^2 x - \tan x = 0$$ for $$-\pi < x < \pi$$. 2. **Rewrite the equation:** Factor the left side:

1. **State the problem:** Show that $$\frac{1-\cos 2\alpha}{\sin 2\alpha} \equiv \tan \alpha.$$\n\n2. **Recall double-angle identities:**\n$$\cos 2\alpha = 1 - 2\sin^2 \alpha$$\n$$

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta)$$. 2. **Rewrite tangent:** Recall that $$\tan(\theta) = \frac{\sin(\theta)}

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta)$$. 2. **Rewrite the terms:** Recall that $$\tan(\theta) = \frac{\sin(\theta

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta) - \tan(\theta)}$$ where $$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$. 2. **Rewrite

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1-\sin(\theta)} - \tan(\theta)$$. 2. **Rewrite the tangent term:** Recall that $$\tan(\theta) = \frac{\sin(\

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1-\sin(\theta)} - \tan(\theta)$$. 2. **Rewrite the tangent term:** Recall that $$\tan(\theta) = \frac{\sin(\

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta) - \tan(\theta)}$$. 2. **Rewrite the tangent function:** Recall that $$\tan(\theta) = \frac{

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta)$$. 2. **Rewrite the tangent term:** Recall that $$\tan(\theta) = \frac{\sin

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta)$$. 2. **Rewrite tangent:** Recall that $$\tan(\theta) = \frac{\sin(\theta)}

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1-\sin(\theta)} - \tan(\theta)$$. 2. **Rewrite the tangent:** Recall that $$\tan(\theta) = \frac{\sin(\theta

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta)$$. 2. **Rewrite the tangent:** Recall that $$\tan(\theta) = \frac{\sin(\the

1. The term \textbf{arctan} refers to the \textbf{inverse tangent function}.\n\n2. It is denoted as $\arctan(x)$ and it gives the angle whose tangent is $x$.\n\n3. More formally, i

1. Problem: Verify if the identity $\csc x \cdot \tan x / \sec x = 1$ is true. Step 1: Express all functions in terms of $\sin x$ and $\cos x$.

1. Problem: Find which expression is equivalent to $\sec^2 x$. Step 1: Recall the Pythagorean identity: $\sec^2 x = 1 + \tan^2 x$.

1. **State the problem:** Given the equation $\sec x - \tan x = 1$, find the value(s) of $x$. 2. **Recall trigonometric identities:** We know that $\sec x = \frac{1}{\cos x}$ and $

1. **State the problem:** Simplify the expression $2\cos^2 x \times \frac{\sin x}{\cos x}$.\n\n2. **Rewrite the expression:** The expression is $2\cos^2 x \cdot \frac{\sin x}{\cos

1. Problem 21: Simplify $$\cos(87\pi + \alpha) \cdot \cos\left(\frac{57\pi}{2} - \alpha\right) \cdot \tan(2025\pi + \alpha)$$. - Use periodicity: $$\cos(87\pi + \alpha) = \cos(\alp

1. The problem is to understand why $\cos(\pi + a)$ equals $-\cos(a)$.\n\n2. Recall the cosine addition formula: $$\cos(x + y) = \cos x \cos y - \sin x \sin y.$$\n\n3. Apply this f