91Ó°ÊÓ

Double angle identities are essential tools in trigonometry. They allow you to express trigonometric functions of twice an angle in terms of functions of the original angle. For example, among the double angle identities, \[ \cos 2x = \cos^2 x - \sin^2 x \] is commonly used. However, this can also be expressed as:

• \(\cos 2x = 2\cos^2 x - 1\)
• \(\cos 2x = 1 - 2\sin^2 x\)

These identities can transform complicated expressions involving double angles into simpler ones. Moreover, for sine, the double angle identity is \(\sin 2x = 2\sin x \cos x\). These identities are especially useful in solving equations, integrating, and simplifying expressions in trigonometric problems. They make it easier to see relationships and simplify calculations.

Cotangent is a fundamental concept in trigonometry. In right triangle trigonometry, cotangent is defined as the ratio of the adjacent side to the opposite side. In terms of sine and cosine, cotangent can be expressed as:\[\cot x = \frac{\cos x}{\sin x}\]It's vital to understand that cotangent is essentially the reciprocal of the tangent function:\[\cot x = \frac{1}{\tan x}\]Understanding how cotangent relates to other trigonometric functions helps in solving equations and proving identities. In our exercise, recognizing the expression \(\frac{\cos x}{\sin x}\) as \(\cot x\) is crucial. This shows a direct application of the cotangent concept in verifying trigonometric identities.

The sine function is one of the primary trigonometric functions, often representing periodic waveforms. It is defined mainly in terms of a right triangle as the ratio of the length of the opposite side to the hypotenuse. From 18 degrees to 0 degrees, the sine value oscillates from 1 to -1 and repeats every 360 degrees. For double angles, the sine of double the angle, \(\sin 2x\), is given by:\[\sin 2x = 2 \sin x \cos x\]This formula is particularly useful for rewriting and simplifying expressions that involve products of sine and cosine. It also comes in handy when dealing with integrals and derivatives in calculus. Recognizing and applying these identities can profoundly simplify complex trigonometric problems and expressions.

The cosine function is another cornerstone of trigonometry. It relates the length of the adjacent side to the hypotenuse in a right triangle. Similar to sine, cosine has a range from 1 to -1 and is periodic every 360 degrees or \(2\pi\) radians. For double angles, cosine can be expressed using several identities; one of them being:\[\cos 2x = 2\cos^2 x - 1\]This identity helps convert expressions involving double angles into quadratics in terms of cosine. In our exercise, knowing this identity allows us to simplify \(1 + \cos 2x\) to \(2\cos^2 x\). This transformation can make verifying identities or solving equations much more manageable, as it simplifies complex expressions involving double angles.