Proving Trigonometric Identities And Finding Real Intervals

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Proving Trigonometric Identities and Finding Real Intervals

Hey math enthusiasts! Let's dive into some cool trigonometry stuff. We're gonna tackle a problem involving trigonometric identities, specifically proving an equation and then figuring out where it actually makes sense (i.e., the real interval). Sounds fun, right?

Proving the Trigonometric Identity: A Step-by-Step Guide

Alright, guys, our mission is to prove that this rather complex-looking equation is true:


tan(180° - x)(1 - cos²x) + cos²x = (sin x - cos x)(1 + sin x cos x) / (-cos x)

Don't worry, it's not as scary as it looks! We'll break it down step by step and use some handy trigonometric identities to simplify things. Remember, the key to these proofs is to start with one side of the equation (usually the more complicated one) and manipulate it until it looks exactly like the other side. Let's get started!

Step 1: Simplify tan(180° - x)

First things first, we need to deal with tan(180° - x). Do you remember the identity related to the tangent function in the second quadrant? We know that tan(180° - x) = -tan(x). This is a crucial first step, as it simplifies the expression significantly. So, let's substitute that in. Our equation now looks like this:


-tan(x)(1 - cos²x) + cos²x = (sin x - cos x)(1 + sin x cos x) / (-cos x)

Step 2: Apply the Pythagorean Identity

Next, let's look at (1 - cos²x). Ring any bells? Yep, it's screaming for the Pythagorean identity: sin²x + cos²x = 1. We can rearrange this to get sin²x = 1 - cos²x. This is super useful! Let's substitute that into our equation:


-tan(x)sin²x + cos²x = (sin x - cos x)(1 + sin x cos x) / (-cos x)

Step 3: Express tan(x) in terms of sin(x) and cos(x)

Now, let's deal with that pesky tan(x). We know that tan(x) = sin(x) / cos(x). Let's plug that in:


-(sin(x) / cos(x)) * sin²x + cos²x = (sin x - cos x)(1 + sin x cos x) / (-cos x)

Step 4: Simplify the Left-Hand Side (LHS)

Let's clean up the left side of the equation. We can multiply the terms:


-sin³x / cos(x) + cos²x = (sin x - cos x)(1 + sin x cos x) / (-cos x)

To combine these terms, we need a common denominator, which is cos(x). So, let's rewrite cos²x as (cos³x) / cos(x):


(-sin³x + cos³x) / cos(x) = (sin x - cos x)(1 + sin x cos x) / (-cos x)

Step 5: Factor the Difference of Cubes (LHS)

On the left side, we have a difference of cubes in the numerator. Remember the factorization formula? a³ - b³ = (a - b)(a² + ab + b²). Let's apply this to -sin³x + cos³x which can be rewritten as cos³x - sin³x:


(cos x - sin x)(cos²x + sin x cos x + sin²x) / cos(x) = (sin x - cos x)(1 + sin x cos x) / (-cos x)

Step 6: Use the Pythagorean Identity Again (LHS)

See that cos²x + sin²x in the numerator? Yep, it's 1, thanks to the Pythagorean identity! Let's simplify that:


(cos x - sin x)(1 + sin x cos x) / cos(x) = (sin x - cos x)(1 + sin x cos x) / (-cos x)

Step 7: Match the Right-Hand Side (RHS)

Now, let's multiply both sides by -1:


-(cos x - sin x)(1 + sin x cos x) / cos(x) = (sin x - cos x)(1 + sin x cos x) / (-cos x)



(sin x - cos x)(1 + sin x cos x) / cos(x) = (sin x - cos x)(1 + sin x cos x) / (-cos x)



(sin x - cos x)(1 + sin x cos x) / -cos(x) = (sin x - cos x)(1 + sin x cos x) / (-cos x)

We did it! The left-hand side now perfectly matches the right-hand side. We have successfully proven the trigonometric identity!

Determining the Real Interval for the Expression

Alright, we've proved the identity. Now, let's figure out where this expression is actually real. This means we need to consider any potential issues, like division by zero or taking the square root of a negative number (though we don't have any square roots here). The main thing to watch out for is division by zero, which happens when the denominator is zero.

The Problem with Cosine

Looking at our proven equation, we see -cos(x) in the denominator. This means the expression will be undefined (and therefore not real) whenever cos(x) = 0. So, we need to find the values of x in the interval [0°, 360°] where cos(x) = 0.

Finding the Zeros of Cosine

Think about the unit circle or the graph of the cosine function. Cosine represents the x-coordinate of a point on the unit circle. It equals zero at 90° and 270°. These are the angles where the x-coordinate is zero. Therefore, the expression is not real at these points.

Determining the Real Interval

So, the expression is real for all values of x in the interval [0°, 360°] except for 90° and 270°. In interval notation, this is:


[0°, 90°) ∪ (90°, 270°) ∪ (270°, 360°]

This means that the expression is real for all angles from 0° up to, but not including, 90°; then from just after 90° up to, but not including, 270°; and finally, from just after 270° up to and including 360°.

Summary of What We've Done

In this math adventure, we've successfully:

  • Proved a complex trigonometric identity using various trigonometric identities and algebraic manipulations.
  • Identified the values of x where the expression is undefined (due to division by zero).
  • Determined the real interval where the expression is valid, excluding the points where the denominator becomes zero.

This shows how important understanding the properties of trigonometric functions and the potential issues that can arise in mathematical expressions is. Keep practicing, and you'll become a trigonometry pro in no time!