Problem 1 - Algebra 2

Question

Verify that each of the following is an identity. $$ \tan \theta(\cot \theta+\tan \theta)=\sec ^{2} \theta $$

Step-by-Step Solution

Answer

The identity \( \tan \theta (\cot \theta + \tan \theta) = \sec^2 \theta \) is verified.

Step 1: Understand the Problem

We need to verify that the given expression on the left side equals the expression on the right side. The identity in question is \( \tan \theta (\cot \theta + \tan \theta) = \sec^2 \theta \).

Step 2: Recall Trigonometric Identities

The basic trigonometric identities we will use include: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \), and \( \sec \theta = \frac{1}{\cos \theta} \), hence \( \sec^2 \theta = 1 + \tan^2 \theta \).

Step 3: Simplify LHS Using Trigonometric Formulas

First, substitute the trigonometric identities: \( \tan \theta (\frac{\cos \theta}{\sin \theta} + \tan \theta) = \tan \theta (\cot \theta + \tan \theta) = \tan \theta \cot \theta + \tan^2 \theta \).

Step 4: Simplify Each Term Separately

Calculate \( \tan \theta \cot \theta = \left( \frac{\sin \theta}{\cos \theta} \right) \left( \frac{\cos \theta}{\sin \theta} \right) = 1 \). Hence, LHS becomes \( 1 + \tan^2 \theta \).

Step 5: Compare with the Right Side

Recall from Step 2 that \( \sec^2 \theta = 1 + \tan^2 \theta \). Therefore, the expression matches the right-hand side identity, verifying our equation.

Key Concepts

Tangent and Cotangent Identities

Tangent (\( \tan \theta \)) and cotangent (\( \cot \theta \)) are trigonometric functions that help us understand the relationships between angles and side lengths in right triangles.
The tangent of an angle \( \theta \) is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle.
Mathematically, this is \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
For cotangent, which is the reciprocal of the tangent, it is the ratio of the length of the adjacent side to the opposite side: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

• The \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) identity means whenever \( \tan \theta \) is used, it can be expressed in terms of sine and cosine.
• Likewise, the \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) identity highlights its relationship with \( \tan \theta \) by showing it as the multiplicative inverse.

Learning these identities is important as they can streamline solving trigonometric problems by transforming complex expressions into simpler forms.

Secant Identity

Secant (\( \sec \theta \)) is another trigonometric function related to cosine. It is defined mathematically as the reciprocal of the cosine function:
\( \sec \theta = \frac{1}{\cos \theta} \).
By understanding this identity, we can transform and simplify expressions by replacing secant with its equivalent in terms of cosine.
A particularly useful form of the secant identity is \( \sec^2 \theta = 1 + \tan^2 \theta \).
This specific identity can often help solve complex trigonometric expressions whenever \( \sec^2 \theta \) appears, it can be substituted to relate back to tangent:

• Understanding that \( \sec^2 \theta = 1 + \tan^2 \theta \) helps connect different trigonometric functions, thereby simplifying equations or proofs that involve multiple functions.

This ability to interchange these identities is essential during trigonometric problem solving, especially when verifying the equality of different expressions.

Verifying Trigonometric Identities

Verifying trigonometric identities is a crucial skill in trigonometry. It involves proving that two different expressions are equivalent, typically by transforming one side of the equation.
In our original exercise, the goal was to prove that \( \tan \theta(\cot \theta + \tan \theta) = \sec^2 \theta \).
This requires:

• Understanding and substituting the trigonometric identities for \( \tan \theta \) and \( \cot \theta \) appropriately into the left side of the expression.
• Transforming expressions by simplifying using algebraic manipulation or known identities, like reducing \( \tan \theta \cot \theta \) to 1.

Successfully verifying an identity means our transformed expression on one side matches the other, such as showing \( 1 + \tan^2 \theta \ = \sec^2 \theta \).Using these techniques deepens the understanding of how trigonometric functions interrelate and enhances problem-solving skills.