📏 trigonometry

1. **State the problem:** Prove that $$\frac{\tan \theta + \sec \theta - 1}{\tan \theta - \sec \theta + 1} = \frac{1 + \sin \theta}{\cos \theta}$$ for an angle $\theta$. 2. **Recal

1. **State the problem:** Use the given trigonometric identities and reduction formulas to solve problems. 2. **Recall the identities:**

1. **Problem Statement:** Arjuna is standing 100 m horizontally from the table. The gift box is initially at height $H$ metres. He shoots an arrow at angle $\theta = 45^\circ$ to b

1. The problem asks to determine the quadrant of the angle $-30^\circ$. 2. Angles are measured counterclockwise from the positive x-axis. Positive angles rotate counterclockwise, n

1. The problem asks to determine the quadrant in which the angle $120^\circ$ lies. 2. Recall the quadrant rules for angles measured in degrees from the positive x-axis:

1. The problem is to understand and simplify the expression $\sin^2(x) \tan^x$. 2. First, clarify the notation: $\sin^2(x)$ means $(\sin(x))^2$, the square of sine of $x$.

1. **State the problem:** Prove or simplify the expression $$\sin^2 x - 3\cos^2 x$$.

1. **Problema:** identificar y resumir las identidades trigonométricas más importantes. 2. Las identidades trigonométricas son igualdades que se cumplen para los ángulos donde las

1. **Planteamos la ecuación dada**. $$1 - 2\cos^2 A + \cos^4 A = \sin^4 A$$

1. **Problem:** Prove that $$4(\cos^2 10^\circ + \sin^2 20^\circ) = 3(\cos 10^\circ + \sin 20^\circ)$$ 2. **Formula and Rules:** Use Pythagorean identity $$\cos^2 x + \sin^2 x = 1$

1. **State the problem:** We are given the function $$y=2\sin x + 1$$ and want to understand its properties. 2. **Formula and explanation:** The function is a sinusoidal function o

1. **Problem Statement:** We are asked to solve problems involving trigonometry combined with solid mensuration, which means calculating lengths, areas, or volumes in 3D shapes usi

1. **State the problem:** We have a right triangle CBA with right angle at B. Given:

1. **State the problem:** We are given a right triangle where the hypotenuse (the slanted side) is $8.63$ m and the angle of inclination at the base is $68^\circ$. We need to find

1. **State the problem:** We want to prove the trigonometric identity $$\sin(x-y) = \sin x \cos y - \cos x \sin y.$$\n\n2. **Recall the sine difference formula:** The sine of a dif

1. **Problem statement:** Given the equation $\csc A + \sec A = \csc B + \sec B$, prove that $$\tan A \tan B = \cot \frac{A + B}{2}$$. 2. **Rewrite the given equation in terms of s

1. **Problem statement:** Given $\csc A + \sec A = \csc B + \sec B$, prove that $\tan A \tan B = \cot \frac{A + B}{2}$. 2. **Rewrite the given equation:**

1. The problem is to find the value of $\tan 30^\circ$. 2. Recall the definition of tangent in a right triangle: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$.

1. The problem is to simplify or understand the expression $\tan a + 1$. 2. Recall that $\tan a$ is the tangent of angle $a$, defined as $\tan a = \frac{\sin a}{\cos a}$.

1. **Problem Statement:** Given a right triangle ABC with angle $\alpha$ where the opposite side length is $8$ units and angle $\theta$ where the adjacent side length is $6$ units,

1. **Problem:** Simplify and prove the identity for (b) $$\frac{\sec80° - 1}{\tan80°} \cdot \frac{\sec40° - 1}{\tan20°}$$. 2. **Formula and rules:** Recall that $$\sec x = \frac{1}