Proof Of Trigonometric Identity $(1 + Sin 2A + Cos 2A)^2 = 4 Cos^2 A (1 + Sin 2A)$

Hey guys! Let's dive into proving this cool trigonometric identity. It looks a bit intimidating at first, but trust me, we'll break it down step by step and it will all make sense. Our goal is to show that $(1 + \sin 2A + \cos 2A)^2$ is indeed equal to $4 \cos^2 A (1 + \sin 2A)$. So, let's roll up our sleeves and get started!

Understanding Trigonometric Identities

Before we jump into the proof, it's super important to have a solid grasp of some fundamental trigonometric identities. These are like the basic building blocks we'll use to manipulate and simplify our expressions. Think of them as your trusty tools in a mathematical toolbox. The identities we'll be using most often are the double angle formulas for sine and cosine, and the Pythagorean identity. These guys are the stars of the show when it comes to simplifying trigonometric expressions, so make sure you're comfortable with them.

• Double Angle Formulas: These formulas are essential for dealing with trigonometric functions of double angles, like $\sin 2A$ and $\cos 2A$. Specifically, we have:
  • $\sin 2A = 2 \sin A \cos A$
  • $\cos 2A = \cos^2 A - \sin^2 A$
  • There are other forms for $\cos 2A$, like $2 \cos^2 A - 1$ and $1 - 2 \sin^2 A$, which can be super handy depending on the situation. For our proof, we'll be using $\cos 2A = 2 \cos^2 A - 1$.
• Pythagorean Identity: This identity is a cornerstone of trigonometry and links sine and cosine together. It states that:
  • $\sin^2 A + \cos^2 A = 1$

These identities are not just formulas to memorize; they're powerful tools that allow us to rewrite trigonometric expressions in different forms. By skillfully applying these identities, we can simplify complex expressions and ultimately prove trigonometric identities. So, keep these close as we move forward in the proof!

Step-by-Step Proof

Okay, let's get down to the nitty-gritty and actually prove the identity. We'll start with the left-hand side (LHS) of the equation, which is $(1 + \sin 2A + \cos 2A)^2$, and we'll manipulate it using our trigonometric identities until it looks exactly like the right-hand side (RHS), which is $4 \cos^2 A (1 + \sin 2A)$. Ready? Let's do this!

1. Start with the Left-Hand Side (LHS)

The left-hand side is $(1 + \sin 2A + \cos 2A)^2$. This is where our journey begins. To tackle this, we'll first rewrite $\sin 2A$ and $\cos 2A$ using their double angle formulas. This will help us break down the expression into smaller, more manageable pieces. So, using the identities we discussed earlier:

• $\sin 2A = 2 \sin A \cos A$
• $\cos 2A = 2 \cos^2 A - 1$

We substitute these into our LHS expression:

$(1 + 2 \sin A \cos A + 2 \cos^2 A - 1)^2$

2. Simplify the Expression

Now, let's simplify things a bit. Notice that we have a +1 and a -1 inside the parentheses? Those guys cancel each other out! This leaves us with:

$(2 \sin A \cos A + 2 \cos^2 A)^2$

Next, we can factor out a common term from inside the parentheses. Both terms have 2 and $\cos A$ in them, so let's pull those out:

$[2 \cos A (\sin A + \cos A)]^2$

3. Square the Expression

Time to square the entire expression. Remember, when you square a product, you square each factor individually. So, we'll square 2, $\cos A, and the (\sin A + \cos A)` term:

$4 \cos^2 A (\sin A + \cos A)^2$

4. Expand the Square

Now, let's expand the squared binomial $(\sin A + \cos A)^2$. Remember the formula for squaring a binomial: $(a + b)^2 = a^2 + 2ab + b^2$. Applying this, we get:

$4 \cos^2 A (\sin^2 A + 2 \sin A \cos A + \cos^2 A)$

5. Use the Pythagorean Identity

Here's where the Pythagorean identity comes to the rescue! We know that $\sin^2 A + \cos^2 A = 1$. So, we can replace $\sin^2 A + \cos^2 A$ in our expression with 1:

$4 \cos^2 A (1 + 2 \sin A \cos A)$

6. Use the Double Angle Formula Again

Look closely! We have $2 \sin A \cos A$ inside the parentheses. That's just $\sin 2A$ in disguise! Let's substitute that back in:

$4 \cos^2 A (1 + \sin 2A)$

7. The Grand Finale

And there you have it! We've successfully transformed the left-hand side into the right-hand side:

$4 \cos^2 A (1 + \sin 2A)$

This is exactly what we wanted to show. We started with $(1 + \sin 2A + \cos 2A)^2$ and, through a series of algebraic manipulations and trigonometric identities, we arrived at $4 \cos^2 A (1 + \sin 2A)$.

Conclusion

In conclusion, we've successfully proven the trigonometric identity $(1 + \sin 2A + \cos 2A)^2 = 4 \cos^2 A (1 + \sin 2A)$. We achieved this by starting with the left-hand side, applying key trigonometric identities such as the double angle formulas and the Pythagorean identity, and simplifying step-by-step until we arrived at the right-hand side. Remember, guys, the key to mastering trigonometric identities is practice and familiarity with the fundamental identities. Keep practicing, and you'll become a trigonometric wizard in no time!

SEO Keywords

To make sure this article gets the attention it deserves, let's sprinkle in some SEO-friendly keywords. These are the terms people might search for when they're looking for help with this kind of problem. We'll make sure they fit naturally into the text, so it still reads smoothly.

• Trigonometric Identity Proof: This is a broad term that captures the essence of what we're doing.
• Prove Trigonometric Identity: A more specific query that targets users looking for guidance on proving identities.
• Double Angle Formula: This is a key concept used in the proof, so it's a valuable keyword.
• Pythagorean Identity: Another crucial identity used in the proof.
• $\sin 2A$ Identity: Specifically targeting the sine double angle formula.
• $\cos 2A$ Identity: Specifically targeting the cosine double angle formula.
• Trigonometry Help: A general term for people seeking assistance with trigonometry.
• Math Proof: A broader category that encompasses mathematical proofs in general.

By including these keywords, we're making it easier for people to find this article when they search online. But remember, the most important thing is to provide clear, helpful, and engaging content. The SEO keywords are just the icing on the cake!

Practice Problems

Now that we've worked through this proof together, why not try your hand at some similar problems? Practice is key to mastering trigonometric identities. Here are a few to get you started:

1. Prove: $(\sin A + \cos A)^2 = 1 + \sin 2A$
2. Prove: $\frac{\sin 2A}{1 + \cos 2A} = \tan A$
3. Prove: $\cos 2A = 2 \cos^2 A - 1$

Work through these problems step-by-step, using the techniques and identities we discussed in this article. If you get stuck, don't worry! Go back and review the steps we took in the main proof. And remember, the more you practice, the more confident you'll become in your ability to tackle trigonometric identities.