Question

Simplify $$ \tan A+\frac{1}{\tan A} $$

Short answer

Answer: The simplified form of the expression \(\tan A + \frac{1}{\tan A}\) is \(\csc A\).

Step-by-step solution

Find a common denominator

To simplify the given expression, first, we need to find a common denominator. In this case, the common denominator is \(\tan A\). So, rewrite the expression with the common denominator \(\tan A\): $$ \tan A + \frac{1}{\tan A} = \frac{\tan^2 A + 1}{\tan A} $$

Use the Pythagorean Identity

We can use the Pythagorean Identity for the tangent function, which is given by $$ \tan^2 A + 1 = \sec^2 A $$ Replace the numerator of the expression with \(\sec^2 A\). $$ \frac{\tan^2 A + 1}{\tan A} = \frac{\sec^2 A}{\tan A} $$

Express secant in terms of cosine

We can express the secant function as the reciprocal of the cosine function, as follows: $$ \sec A = \frac{1}{\cos A} $$ Replace the \(\sec\) function in the expression with \(\frac{1}{\cos}\). $$ \frac{\sec^2 A}{\tan A} = \frac{\left(\frac{1}{\cos A}\right)^2}{\tan A} $$

Express tangent in terms of sine and cosine

Next, we express the tangent function as the ratio of the sine and cosine functions, as follows: $$ \tan A = \frac{\sin A}{\cos A} $$ Replace the \(\tan\) function with the quotient of the sine and cosine functions. $$ \frac{\left(\frac{1}{\cos A}\right)^2}{\frac{\sin A}{\cos A}} = \frac{\cos^{-2} A}{\frac{\sin A}{\cos A}} $$

Simplify the expression

Finally, simplify the expression by inverting and multiplying. $$ \frac{\cos^{-2} A}{\frac{\sin A}{\cos A}} = \frac{1}{\cos^2 A} \cdot \frac{\cos A}{\sin A} = \frac{\cos A}{\sin A \cos A} = \frac{1}{\sin A} $$ The simplified expression is \(\frac{1}{\sin A}\), which is equivalent to \(\csc A\). Therefore, the simplified expression is $$ \tan A+\frac{1}{\tan A} = \csc A $$

Key concepts

tan function simplification

When dealing with trigonometric expressions, simplification is often required to make calculations or understand the expressions more easily. The tangent function, denoted as \( \tan A \), is one such function that sometimes needs simplification, especially in more complex expressions.
Understanding how to simplify \( \tan A + \frac{1}{\tan A} \) is essential.To begin, it's useful to find a common denominator, which in this scenario is \( \tan A \). This leads to the expression \( \frac{\tan^2 A + 1}{\tan A} \), providing a clearer form to work with. Remember:

• Common denominators help to combine fractions into one manageable piece.
• This step makes it easier to apply further identities and simplifications later on.

Simplifying expressions like these makes them less convoluted, as well as sets a foundation for applying other identities efficiently.

Pythagorean identity

The Pythagorean Identity is central to simplifying trigonometric expressions. One variant of this identity for the tangent function is \( \tan^2 A + 1 = \sec^2 A \). This identity allows us to replace \( \tan^2 A + 1 \) with \( \sec^2 A \) in our expression.The identity itself comes from the relationship between sine and cosine, specifically:

• The Pythagorean Identity for sine and cosine: \( \sin^2 A + \cos^2 A = 1 \).
• Derived forms use the relationship: \( \tan A = \frac{\sin A}{\cos A} \) and \( \sec A = \frac{1}{\cos A} \).

Using the Pythagorean Identity transforms \( \frac{\tan^2 A + 1}{\tan A} \) into \( \frac{\sec^2 A}{\tan A} \), pushing the expression closer to a simpler form. It's these identities that make transformations and simplifications in trigonometry possible and practical.

trigonometric function relationships

Understanding the relationships between various trigonometric functions is key in simplifying expressions. Functions like tangent and secant are often expressed in terms of sine and cosine, which can greatly simplify calculations or derivations.In our exercise, \( \sec A \) was expressed as \( \frac{1}{\cos A} \) and \( \tan A \) as \( \frac{\sin A}{\cos A} \). These expressions help to simplify the exercise even further:

• Rewriting \( \sec^2 A \): \( \left(\frac{1}{\cos A}\right)^2 = \frac{1}{\cos^2 A} \).
• Substituting \( \tan A \): Turns the expression into dividing by the sine-cosine ratio.

Through these relationships, the original complex expression \( \tan A + \frac{1}{\tan A} \) simplifies neatly to \( \csc A \) or \( \frac{1}{\sin A} \), demonstrating the utility of knowing how to manipulate trigonometric function relationships for easier problem solving.