📏 trigonometry

1. **Prove that** $\tan x + \cot x = \frac{2}{\sin 2x}$. - Start with the left-hand side (LHS):

1. The problem is to simplify the expression $ (1 + \tan^2(x)) \sec^2(x) \, dx $.\n\n2. Recall the Pythagorean identity: $$ 1 + \tan^2(x) = \sec^2(x) $$\nThis identity is fundament

1. **Problem statement:** Find $\sin(\beta)$ in the right triangle with sides $BC=8$, $CA=15$, and hypotenuse $AB=17$, where $\beta$ is the angle at vertex $B$. 2. **Recall the sin

1. **Problem statement:** Find $\tan(\alpha)$ in a right triangle where the side opposite $\alpha$ is 21, the side adjacent to $\alpha$ is 20, and the hypotenuse is 29. 2. **Formul

1. **Problem Statement:** Find $\cos(\beta)$ in a right triangle where the side opposite to angle $\beta$ is 3, the side adjacent to $\beta$ is 4, and the hypotenuse is 5. 2. **For

1. **Problem statement:** Find $\cos(\beta)$ in the right triangle with sides $BC=5$, $AC=12$, and hypotenuse $AB=13$, where $\beta$ is the angle at vertex $B$. 2. **Formula:** In

1. **Problem Statement:** Find $\sin(\beta)$ in a right triangle with sides 3, 4, and 5, where the right angle is at vertex C and angle $\beta$ is at vertex B. 2. **Recall the defi

1. **Problem statement:** We have a right triangle ABC with a right angle at C. The hypotenuse AB is 13, side BC is 12, and side AC is 5.

1. **Problem statement:** Find $\cos(\alpha)$ in the right triangle with sides $BC=7$, $AC=24$, and hypotenuse $AB=25$, where $\alpha$ is the angle at vertex $A$. 2. **Formula:** I

1. **Problem statement:** Find $\sin(\alpha)$ in the given right triangle where $\alpha$ is the angle at vertex A. 2. **Recall the definition of sine in a right triangle:**

1. **Problem 5:** Prove the identity $$\frac{\sin^2 A}{1 - \cos^4 A} + \frac{\cos^2 A}{1 - \sin^4 A} \equiv \frac{3}{2 + \sin^2 A \cos^2 A}$$ 2. **Step 1:** Recognize that $$1 - \c

1. **Problem:** Prove the identity $$\frac{1}{(\sec A - \tan A)} - \frac{1}{\cos A} \equiv \frac{1}{\cos A} - \frac{1}{(\sec A + \tan A)}$$

1. **Problem:** Prove that $$\frac{\cos^2\theta - \sin^2\theta}{(1 - \tan^2\theta) \sin^2\theta} \equiv \cot^2\theta$$. **Step 1:** Recall the identities:

1. **State the problem:** Given that $\tan\theta + \cot\theta = 2$, find the value of $\tan^3\theta + \cot^3\theta$. 2. **Recall the formula:** The sum of cubes formula is:

1. Problem 4: Prove that $$\frac{\sin^2 \theta}{1 + \cot \theta} + \frac{\cos^2 \theta}{1 + \tan \theta} \equiv 1 - \sin \theta \cos \theta$$. 2. Use the identities: $$\tan \theta

1. **Problem:** Prove the identity $$2(\sin^6 A + \cos^6 A) - 3(\sin^4 A + \cos^4 A) + 1 \equiv 0$$ **Step 1:** Use the identity for sum of powers: $$a^3 + b^3 = (a+b)^3 - 3ab(a+b)

1. **State the problem:** Prove the trigonometric identity $$\frac{1}{2} \left( \cot x + \tan x \right) = \csc 2x$$

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1-\sin(\theta)} - \tan(\theta)$$. 2. **Recall formulas and identities:**

1. **Problem statement:** Prove the trigonometric identities: (a) $\tan \theta \sin \theta + \cos \theta = \sec \theta$

1. **Problem statement:** Prove the trigonometric identities: (a) $\tan \theta \sin \theta + \cos \theta = \sec \theta$

1. Convert from degrees to radians. The formula to convert degrees to radians is: