Problem 38 - Contemporary Precalculus

Question

State whether or not the equation is an identity. If it is an identity, prove it. $$\cot ^{2} x-1=\csc ^{2} x$$

Step-by-Step Solution

Answer

Answer: Yes, the given equation is an identity involving trigonometric functions. It is a transformation of the Pythagorean Identity involving cotangent and cosecant, $\cot^2 x + 1 = \csc^2 x$. The equation is true for all values of \(x\) within the domain of cotangent and cosecant functions.

Step 1: Recall the Pythagorean Trigonometric Identity

Recall the Pythagorean Trigonometric Identity involving cotangent and cosecant: $$\cot^2 x + 1 = \csc^2 x$$

Step 2: Compare the Given Equation to the Pythagorean Identity

Compare the given equation: $$\cot ^{2} x-1=\csc ^{2} x$$ to the Pythagorean Identity: $$\cot^2 x + 1 = \csc^2 x$$

Step 3: Determine if the Equation is an Identity

Notice that if we add 1 to both sides of the given equation, it becomes the Pythagorean Identity: $$\cot ^{2} x-1 + 1 = \csc ^{2} x + 1$$ $$\cot^2 x = \csc^2 x - 1$$ Adding 1 to both sides, we get: $$\cot^2 x + 1 = \csc^2 x$$ Since the Pythagorean Identity holds true for all values of \(x\) within the domain of cotangent and cosecant, the given equation is just the Pythagorean Identity transformed by subtracting 1 from both sides. Thus, the given equation is an identity as well.

Step 4: Prove the Identity

To prove the identity, we can start with the Pythagorean Identity: $$\cot^2 x + 1 = \csc^2 x$$ Subtract 1 from both sides of the equation: $$\cot^2 x = \csc^2 x - 1$$ Now, add 1 to both sides of the equation: $$\cot ^{2} x-1=\csc ^{2} x$$ This confirms that the given equation is indeed an identity, and we've shown the transformation from one form to another.

Key Concepts

Pythagorean Identity

The Pythagorean identity is one of the key building blocks in trigonometry, just like the Pythagorean theorem is for geometry. It helps us understand the relationship between different trigonometric functions. In this context, we focus on the identity that involves cotangent and cosecant:

• \( \cot^2 x + 1 = \csc^2 x \)

This identity is essential for proving many other trigonometric identities, as it establishes a link between these two trigonometric terms. The equation tells us that when you add 1 to the square of cotangent, you get the square of cosecant.
It's critical to be familiar with manipulating this identity to transform equations and demonstrate that they hold true for various angles. By mastering this, you'll be equipped to handle more complex trigonometric equations and identities in trigonometry.

Cotangent

Cotangent (\(\cot x\)) is one of the less common trigonometric functions, but it is equally important as sine and cosine. It is the reciprocal of tangent, which means:

• \( \cot x = \frac{1}{\tan x} \)

Alternatively, this can also be expressed using sine and cosine:

• \( \cot x = \frac{\cos x}{\sin x} \)

Cotangent is particularly useful in solving trigonometric equations and proving identities because it can help transform expressions into alternate forms.
Understanding cotangent becomes crucial when working with identities like the Pythagorean identity. It allows us to see how cotangent fits with the other trigonometric functions, linking them through relationships like the one shown in the identity.
The domain of cotangent is all real numbers except for multiples of \(\pi\), due to the points at which sine equals zero, causing cotangent to become undefined. Recognizing its domain helps ensure we apply trigonometric identities correctly.

Cosecant

Cosecant (\(\csc x\)) is the reciprocal of sine. It is an essential function in trigonometry for extending relationships between functions, similar to secant and tangent. The basic fact about cosecant is:

• \( \csc x = \frac{1}{\sin x} \)

This reciprocal relationship means anywhere sine is zero, cosecant becomes undefined, specifically at integer multiples of \(\pi\).
In the Pythagorean identity \(\cot^2 x + 1 = \csc^2 x\), cosecant's role is clear. It shows how increasing functions (squared, here) relate to each other across an identity framework.
Cosecant isn't often used on its own but is crucial when diving into manipulations and transformations in other trigonometric expressions and equations. By comprehending how cosecant interacts with other functions, you can better navigate complex trigonometric proofs and identities.