Express $4 \cos^2(165^{\circ})-2$ As A Single Trigonometric Function

Hey guys! Today, we're diving deep into the fascinating world of trigonometry to tackle a seemingly complex expression: $4 \cos^2(165^\circ) - 2$. Our mission? To simplify this and express it as a single, elegant trigonometric function. Buckle up, because we're about to embark on a mathematical journey filled with identities, transformations, and a whole lot of cosine magic!

Understanding the Challenge

At first glance, $4 \cos^2(165^\circ) - 2$ might look a bit intimidating. We've got a cosine squared, a specific angle (165 degrees), and some arithmetic operations thrown in for good measure. But don't worry, we're not going to let this expression defeat us! The key to solving this lies in recognizing that this expression bears a striking resemblance to a well-known trigonometric identity – the double-angle formula for cosine. This is where our adventure truly begins.

The Double-Angle Formula: Our Secret Weapon

The double-angle formula for cosine comes in a few different flavors, but the one that's particularly relevant to our problem is:

$\cos(2x) = 2\cos^2(x) - 1$

Notice how the right side of this equation, $2\cos^2(x) - 1$, looks suspiciously similar to our expression, $4 \cos^2(165^\circ) - 2$. This is no coincidence! By cleverly manipulating our expression and applying this identity, we can significantly simplify it. This is a crucial step in our journey, as it allows us to transform the expression into a form that is easier to work with and ultimately express as a single trigonometric function. Remember, the beauty of mathematics lies in recognizing patterns and using the right tools to unravel complexity.

Manipulating the Expression: Setting the Stage

Our expression is $4 \cos^2(165^\circ) - 2$. To make it look exactly like the double-angle formula, we need to do a little bit of algebraic maneuvering. First, let's factor out a 2 from the entire expression:

$4 \cos^2(165^\circ) - 2 = 2(2 \cos^2(165^\circ) - 1)$

Now, the expression inside the parentheses, $2 \cos^2(165^\circ) - 1$, is a perfect match for the right side of the double-angle formula! This is a pivotal moment in our simplification process. We've successfully transformed the original expression into a form where we can directly apply the double-angle identity, bringing us closer to our goal of expressing it as a single trigonometric function.

Applying the Double-Angle Formula

With our expression neatly massaged into the form $2(2 \cos^2(165^\circ) - 1)$, we can now confidently apply the double-angle formula for cosine. Recall that the formula is:

$\cos(2x) = 2\cos^2(x) - 1$

In our case, $x = 165^\circ$. So, we can substitute this into the formula:

$2 \cos^2(165^\circ) - 1 = \cos(2 \cdot 165^\circ) = \cos(330^\circ)$

This substitution is a game-changer! We've effectively replaced a complex term involving cosine squared with a single cosine function of a doubled angle. This is a significant leap towards simplification and demonstrates the power of trigonometric identities in transforming expressions. The double-angle formula acts as a bridge, connecting seemingly different forms of trigonometric expressions and allowing us to navigate towards simpler representations.

The Simplified Expression: A Step Closer to the Finish Line

Now, let's plug this back into our original expression:

$2(2 \cos^2(165^\circ) - 1) = 2 \cos(330^\circ)$

We've made excellent progress! We've transformed the initial expression into something much simpler: $2 \cos(330^\circ)$. However, we're not quite at the finish line yet. We need to determine the value of $\cos(330^\circ)$ to fully express our answer as a single trigonometric function. This requires us to delve into the unit circle and understand how cosine behaves in different quadrants.

Evaluating $\cos(330^\circ)$

To find the value of $\cos(330^\circ)$, we'll use our knowledge of the unit circle and reference angles. The unit circle is an invaluable tool in trigonometry, providing a visual representation of trigonometric functions for various angles. It allows us to easily determine the sine, cosine, and tangent of angles, especially those that are multiples of 30 and 45 degrees.

Reference Angles: Navigating the Unit Circle

An angle of $330^\circ$ lies in the fourth quadrant. To find its reference angle, we subtract it from $360^\circ$:

$360^\circ - 330^\circ = 30^\circ$

The reference angle, $30^\circ$, is the acute angle formed between the terminal side of our angle ($330^\circ$) and the x-axis. It allows us to relate the trigonometric functions of $330^\circ$ to the trigonometric functions of the familiar $30^\circ$ angle. This concept of reference angles is fundamental in trigonometry, enabling us to simplify calculations and understand the behavior of trigonometric functions across different quadrants.

Cosine in the Fourth Quadrant: Positive Vibes

In the fourth quadrant, cosine is positive. This is a crucial piece of information, as it tells us the sign of $\cos(330^\circ)$. Remembering the mnemonic