Question

Simplify. $$\frac{\sin ^{2} x+\cos ^{2} x}{1-\cos ^{2} x}$$

Short answer

\(\csc^2x\)

Step-by-step solution

Apply the Pythagorean Identity

Recognize that the numerator \(\sin^2x + \cos^2x\) is a Pythagorean Identity, which is equal to 1.

Simplify the Denominator

Observe that \(1 - \cos^2x\) is another form of \(\sin^2x\), because \(\sin^2x + \cos^2x = 1\) implies \(\sin^2x = 1 - \cos^2x\).

Divide Numerator by Denominator

Divide the numerator, which is 1, by the denominator, now simplified to \(\sin^2x\), to get \(\frac{1}{\sin^2x}\).

Convert to Cosecant

Use the definition of cosecant, which is \(\csc(x) = \frac{1}{\sin(x)}\), to express \(\frac{1}{\sin^2x}\) as \(\csc^2x\).

Key concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics that provide relationships between various trigonometric functions. They allow us to rewrite expressions and equations in a simpler or more useful form. One of the most important of these is the Pythagorean Identity, which states that for any angle x, \(\sin^2x + \cos^2x = 1\). This identity is derived from the Pythagorean Theorem applied to a right triangle, where the sine and cosine represent the legs and the hypotenuse is of unit length.

These identities not only help in simplifying expressions but also in solving complex trigonometric equations. By using the Pythagorean Identity, as seen in the original exercise, we reduce what may seem like a unique problem into a form that's often much easier to manage.

Simplify Trigonometric Expressions

To simplify trigonometric expressions, mathematicians often rely on a set of strategies powered by the aforementioned identities. The core of simplifying trigonometric expressions is to express them in the simplest form possible, often to facilitate easier computation or to better analyze their characteristics. For instance, recognizing that \(\sin^2x + \cos^2x\) simplifies to 1 and that \(1 - \cos^2x\) converts to \(\sin^2x\) are acts of simplification using trigonometric identities.

Once these identities are applied, we can manipulate the expressions to reveal new forms that might be more advantageous for our purpose, like finding the value of an angle or solving for x. Simplification also often involves converting a complex expression into a basic trigonometric function, like turning \(\frac{1}{\sin^2x}\) into \(\csc^2x\), making both evaluation and comprehension much smoother.

Cosecant Function

The cosecant function, often written as \(\csc(x)\), is one of the six primary trigonometric functions. It is defined as the reciprocal of the sine function, that is \(\csc(x) = \frac{1}{\sin(x)}\). While not as commonly used as the sine or cosine functions, it plays a critical role in various mathematical scenarios, particularly in simplifications like the one in our example problem.

If \(\sin(x)\) represents the ratio of the opposite side to the hypotenuse in a right triangle, then \(\csc(x)\) represents the ratio of the hypotenuse to the opposite side. This reciprocal relationship often arises in problems involving waves, oscillations, and other periodic phenomena. By converting to the cosecant function, we can transform a complex expression into a simpler one, thereby making our mathematical journey smoother.