Problem 50 - Algebra 2

Question

Verify each identity. $$ (\cot \theta+1)^{2}=\csc ^{2} \theta+2 \cot \theta $$

Step-by-Step Solution

Answer

The given identity (\(\cot \theta+1)^{2}=\csc ^{2} \theta+2 \cot \theta\) is verified.

Step 1: Expand the left-hand side

First, expand the left-hand side of the equation (\(\cot \theta + 1)^2 = \cot^2 \theta + 2\cot \theta + 1\).

Step 2: Replace \(\cot^2 \theta\) by \(\csc^2 \theta - 1\)

By using the Pythagorean trigonometric identity, replace \(\cot^2 \theta\) by \(\csc^2 \theta - 1\). So, the equation becomes \(\csc^2 \theta + 2\cot \theta\). This is the right-hand side of the original equation.

Step 3: Conclusion

After simplifying and unifying both sides in terms of \(\csc \theta\) and \(\cot \theta\), we can see that both sides are identical, therefore the given identity is verified.

Key Concepts

Cotangent Identity

The cotangent identity helps us express the cotangent function, usually denoted as \( \cot \theta \), in relation to sine and cosine. Cotangent is the reciprocal of the tangent function. In terms of sine and cosine, it is expressed as:

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

This relationship is useful for converting expressions involving tangent into forms using only sine and cosine.
Additionally, cotangent squared \( (\cot^2 \theta) \) frequently appears in trigonometric identities and equations alongside the other main trigonometric functions. In the provided exercise, understanding cotangent's definition is crucial for manipulating the identity and simplifying terms.

Cosecant Identity

Cosecant is another fundamental trigonometric function, represented as \( \csc \theta \). It is the reciprocal of the sine function:

• \( \csc \theta = \frac{1}{\sin \theta} \)

Due to its relationship with sine, cosecant is often used in more complex trigonometric identities and expressions.
When you square the cosecant, \( \csc^2 \theta \), you are effectively dealing with the square of one over sine. This makes it a key player in various identities, such as the Pythagorean identities.
In the given exercise, replacing terms involving cotangent or modifying expressions can often involve using the cosecant identity to make the equation consistent on both sides.

Pythagorean Identity

Pythagorean identities are some of the most widely used in trigonometry, enabling the transformation between different trigonometric functions. One such identity involves cotangent and cosecant:

• \( \cot^2 \theta + 1 = \csc^2 \theta \)

These identities derive from the Pythagorean theorem, relating the squares of sine and cosine to 1. Here, it illustrates the relationship between \( \cot^2 \theta \) and \( \csc \theta \).
In the context of the exercise, replacing \( \cot^2 \theta \) with \( \csc^2 \theta - 1 \) is a direct application of this Pythagorean identity.
This swap helps in transforming the equation into a simpler form, proving that both sides of the given identity are equal. Recognizing and knowing how to apply this identity is essential when dealing with trigonometric proofs and verifications.