Simplify Trig Expressions: Fundamental Identities Guide

Are you struggling with simplifying trigonometric expressions? Don't worry, you're not alone! Trigonometry can seem daunting at first, but with a solid understanding of fundamental identities, you can simplify even the most complex expressions. This guide will walk you through how to use these identities to solve the expressions you provided. Let's dive in!

Understanding Fundamental Trigonometric Identities

Before we tackle the problems, it's crucial to understand the fundamental trigonometric identities. These are the basic equations that relate the trigonometric functions to each other. Mastering these identities is the key to simplifying expressions and solving trigonometric equations. Here’s a rundown of some of the most important ones:

• Reciprocal Identities: These identities define the reciprocal relationships between trigonometric functions:
  • $\csc(x) = \frac{1}{\sin(x)}$
  • $\sec(x) = \frac{1}{\cos(x)}$
  • $\cot(x) = \frac{1}{\tan(x)}$
• Quotient Identities: These identities relate tangent and cotangent to sine and cosine:
  • $\tan(x) = \frac{\sin(x)}{\cos(x)}$
  • $\cot(x) = \frac{\cos(x)}{\sin(x)}$
• Pythagorean Identities: These identities are derived from the Pythagorean theorem and are incredibly useful:
  • $\sin^2(x) + \cos^2(x) = 1$
  • $1 + \tan^2(x) = \sec^2(x)$
  • $1 + \cot^2(x) = \csc^2(x)$

These identities are the tools we'll use to simplify the given expressions. Remember, the goal is to rewrite the expressions in a simpler form, often by canceling terms or using substitutions based on these identities. By understanding and applying these identities, you can transform complex trigonometric expressions into more manageable forms. Let's move on to solving the expressions step by step!

Simplifying Trigonometric Expressions: A Step-by-Step Guide

Now that we've reviewed the fundamental identities, let's apply them to the expressions you provided. We'll break down each expression step by step, showing you how to simplify it using the appropriate identities. Remember, practice is key, so feel free to try these on your own first and then compare your solutions with our explanations.

1. $\frac{\sin (x) \tan (x)}{\cos (x)}=$ ?

To simplify this expression, we can start by substituting the definition of $\tan(x)$ from the quotient identities. We know that $\tan(x) = \frac{\sin(x)}{\cos(x)}$, so let's replace $\tan(x)$ in the expression:

$\frac{\sin (x) \tan (x)}{\cos (x)} = \frac{\sin (x) \cdot \frac{\sin(x)}{\cos(x)}}{\cos (x)}$

Now we have a fraction within a fraction, which can be simplified by multiplying the numerators together:

$= \frac{\sin^2(x)}{\cos^2(x)}$

Notice that we now have $\frac{\sin^2(x)}{\cos^2(x)}$. This looks very similar to another fundamental identity! Recall that $\tan(x) = \frac{\sin(x)}{\cos(x)}$. Therefore, $\tan^2(x) = \frac{\sin^2(x)}{\cos^2(x)}$. So, we can simplify the expression further:

$= \tan^2(x)$

So, the simplified form of the expression $\frac{\sin (x) \tan (x)}{\cos (x)}$ is $\tan^2(x)$. This example demonstrates how substituting fundamental identities can transform a seemingly complex expression into a much simpler form. Let's move on to the next expression!

2. $\tan (x) \cos (x)=$ ?

In this case, we again encounter $\tan(x)$, which we know can be expressed in terms of sine and cosine using the quotient identities. Let's substitute $\tan(x)$ with $\frac{\sin(x)}{\cos(x)}$:

$\tan (x) \cos (x) = \frac{\sin(x)}{\cos(x)} \cdot \cos(x)$

Now, we have $\cos(x)$ in both the numerator and the denominator. These terms can cancel each other out:

$= \sin(x)$

So, the simplified form of the expression $\tan (x) \cos (x)$ is simply $\sin(x)$. This example illustrates the power of canceling terms after making appropriate substitutions using the fundamental identities. It often leads to significant simplification.

3. $\sin (x) \cos (x)=$ ?

This expression, $\sin (x) \cos (x)$, is already in a fairly simple form. There isn't a direct application of the basic trigonometric identities that will simplify it further into a single trigonometric function. However, it's worth noting that this expression appears in the double-angle identity for sine, which is $\sin(2x) = 2\sin(x)\cos(x)$. If we wanted to rewrite it in terms of $\sin(2x)$, we could multiply the expression by $\frac{2}{2}$:

$\sin(x)\cos(x) = \frac{1}{2} \cdot 2 \sin(x)\cos(x)$

Now, we can substitute $2\sin(x)\cos(x)$ with $\sin(2x)$:

$= \frac{1}{2} \sin(2x)$

While the original expression $\sin (x) \cos (x)$ is already quite simple, this transformation demonstrates how identities can be used to rewrite expressions in different forms. Depending on the context, this might be a useful simplification. However, for the purpose of basic simplification, leaving it as $\sin(x)\cos(x)$ is perfectly acceptable.

4. $(\sec (x))^2-1=$ ?

For this expression, we can turn to the Pythagorean identities. Recall the identity $1 + \tan^2(x) = \sec^2(x)$. We can rearrange this identity to match the form of our expression:

$\sec^2(x) - 1 = \tan^2(x)$

Therefore, $(\sec (x))^2-1$ simplifies directly to $\tan^2(x)$. This is a straightforward application of the Pythagorean identities, showing how they can be used to quickly simplify expressions involving squares of trigonometric functions.

5. $(\tan (x))^2+\sin (x) \csc (x)=$ ?

This expression combines different trigonometric functions, so we'll need to use a combination of identities to simplify it. First, let's address the $\csc(x)$ term. Recall the reciprocal identity $\csc(x) = \frac{1}{\sin(x)}$. We can substitute this into our expression:

$\tan^2(x) + \sin(x) \csc(x) = \tan^2(x) + \sin(x) \cdot \frac{1}{\sin(x)}$

Now, we have $\sin(x)$ in both the numerator and denominator, which can cancel each other out:

$= \tan^2(x) + 1$

Notice that we now have $\tan^2(x) + 1$. This is another Pythagorean identity in disguise! Recall the identity $1 + \tan^2(x) = \sec^2(x)$. So, we can simplify the expression further:

$= \sec^2(x)$

Thus, the simplified form of the expression $(\tan (x))^2+\sin (x) \csc (x)$ is $\sec^2(x)$. This final example illustrates how combining reciprocal and Pythagorean identities can lead to a concise and simplified result.

Conclusion: Mastering Trigonometric Identities

Simplifying trigonometric expressions might seem challenging at first, but by mastering the fundamental identities and practicing their application, you can confidently tackle these problems. Remember to:

• Know your identities: Reciprocal, quotient, and Pythagorean identities are your primary tools.
• Look for substitutions: Identify terms that can be replaced using identities.
• Simplify and cancel: After substitutions, simplify the expression and cancel out common terms.
• Practice regularly: The more you practice, the more comfortable you'll become with these simplifications.

By following these steps, you'll be well on your way to mastering trigonometric simplifications! For further exploration and practice, consider visiting Khan Academy's Trigonometry section. Happy simplifying!