Trigonometric Expression Evaluation: A Math Challenge

Hey math whizzes! Today, we've got a seriously cool trigonometric expression that's going to put your knowledge to the test. We're diving deep into the world of cosines to evaluate the following: (cos 64° × cos 4° - cos 86° × cos 26°) ÷ (cos 71° × cos 3° + cos 87° × cos 19°). And guess what? We've got some juicy multiple-choice options for you: A. 0, B. 1, C. 5, D. 8. Can you crack this code and find the correct answer? Let's get those brains buzzing!

Unpacking the Numerator: The Power of Cosine Product-to-Sum

Alright guys, let's first tackle the top part of our expression, the numerator: cos 64° × cos 4° - cos 86° × cos 26°. To make this beast more manageable, we're going to bring in some trusty trigonometric identities. Specifically, the product-to-sum identities are going to be our best friends here. Remember these gems? They allow us to transform products of sines and cosines into sums or differences of cosines.

The first product we'll work with is cos 64° × cos 4°. Using the identity cos A × cos B = 1/2 [cos(A - B) + cos(A + B)], we can rewrite this as:

1/2 [cos(64° - 4°) + cos(64° + 4°)]

Which simplifies to:

1/2 [cos 60° + cos 68°]

Now, let's look at the second product in the numerator: cos 86° × cos 26°. Applying the same identity, we get:

1/2 [cos(86° - 26°) + cos(86° + 26°)]

This simplifies to:

1/2 [cos 60° + cos 112°]

So, the entire numerator becomes:

1/2 [cos 60° + cos 68°] - 1/2 [cos 60° + cos 112°]

Let's factor out the 1/2:

1/2 [ (cos 60° + cos 68°) - (cos 60° + cos 112°) ]

1/2 [ cos 60° + cos 68° - cos 60° - cos 112° ]

Notice how the cos 60° terms cancel each other out! That's a good sign, meaning we're heading in the right direction. We're left with:

1/2 [ cos 68° - cos 112° ]

Now, we can use another handy identity: the sum-to-product identity for cosines, in the form cos A - cos B = -2 sin[(A + B)/2] sin[(A - B)/2]. Let A = 68° and B = 112°.

Applying this, we get:

1/2 [ -2 sin[(68° + 112°)/2] sin[(68° - 112°)/2] ]

1/2 [ -2 sin(180°/2) sin(-44°/2) ]

1/2 [ -2 sin(90°) sin(-22°) ]

We know that sin 90° = 1. Also, sin(-x) = -sin(x), so sin(-22°) = -sin(22°).

Substituting these values:

1/2 [ -2 * 1 * (-sin 22°) ]

1/2 [ 2 sin 22° ]

This further simplifies to sin 22°.

So, the entire numerator boils down to sin 22°. Pretty neat, right? We’ve successfully simplified a complex expression into a single sine term using a couple of fundamental trigonometric identities. Keep this result handy, because we're about to tackle the denominator, and the magic is far from over!

Decoding the Denominator: More Trigonometric Gymnastics

Now, let's shift our focus to the denominator: cos 71° × cos 3° + cos 87° × cos 19°. Just like with the numerator, we'll lean on our trusty product-to-sum identities to break this down. We have two products to convert:

First, let's look at cos 71° × cos 3°. Using the identity cos A × cos B = 1/2 [cos(A - B) + cos(A + B)]:

1/2 [cos(71° - 3°) + cos(71° + 3°)]

Which simplifies to:

1/2 [cos 68° + cos 74°]

Next, we have cos 87° × cos 19°. Applying the same identity:

1/2 [cos(87° - 19°) + cos(87° + 19°)]

This simplifies to:

1/2 [cos 68° + cos 106°]

Now, let's put these back into the denominator expression:

1/2 [cos 68° + cos 74°] + 1/2 [cos 68° + cos 106°]

Factor out the 1/2:

1/2 [ (cos 68° + cos 74°) + (cos 68° + cos 106°) ]

1/2 [ cos 68° + cos 74° + cos 68° + cos 106° ]

Combine the like terms:

1/2 [ 2 cos 68° + cos 74° + cos 106° ]

This looks a bit more complicated, doesn't it? But wait, we can use a clever trick with complementary angles and supplementary angles. Remember that cos(180° - x) = -cos x? Let's apply this to cos 106°:

cos 106° = cos(180° - 74°) = -cos 74°

Now, substitute this back into our denominator expression:

1/2 [ 2 cos 68° + cos 74° + (-cos 74°) ]

1/2 [ 2 cos 68° + cos 74° - cos 74° ]

Voila! The cos 74° terms cancel out, leaving us with:

1/2 [ 2 cos 68° ]

Which simplifies beautifully to cos 68°.

So, the denominator is cos 68°. We've managed to simplify both the numerator and the denominator using a combination of product-to-sum and angle properties. The journey was packed with identities, but the result is much cleaner. Now, the final step is to put these simplified parts together and see what the actual value of the entire expression is. Get ready for the grand finale!

The Grand Finale: Assembling the Pieces and Finding the Answer

We’ve done the heavy lifting, guys! We’ve simplified the numerator to sin 22° and the denominator to cos 68°. Now, it's time to put them together and evaluate the original expression:

(cos 64° × cos 4° - cos 86° × cos 26°) ÷ (cos 71° × cos 3° + cos 87° × cos 19°)

This is equivalent to:

sin 22° ÷ cos 68°

At first glance, it might seem like we're stuck, as sin and cos are different. But remember our good old friend, the complementary angle identity! This states that sin x = cos (90° - x) and cos x = sin (90° - x). Let's use this to our advantage.

We can rewrite sin 22° using this identity:

sin 22° = cos (90° - 22°)

sin 22° = cos 68°

Now, substitute this back into our division:

cos 68° ÷ cos 68°

And what do we get when we divide a number by itself? That's right, 1!

So, the value of the entire expression is 1.

Let's recap the journey: we started with a complex fraction involving products of cosines. We used the product-to-sum identities to break down these products into sums and differences. Then, we employed the sum-to-product identity and properties of supplementary angles in the numerator to simplify it to sin 22°. In the denominator, we again used product-to-sum and supplementary angle properties to simplify it to cos 68°. Finally, using the complementary angle identity, we showed that sin 22° is equal to cos 68°, leading to the final answer of 1.

This problem beautifully illustrates the power and elegance of trigonometric identities. By systematically applying these rules, we can transform complicated expressions into much simpler forms. It’s like solving a puzzle, where each identity is a key that unlocks a new level of simplification. So, when faced with a daunting trigonometric expression, don't panic! Just remember your identities, break it down step-by-step, and you’ll often find a surprisingly simple solution waiting for you.

The Verdict: Which Option is Correct?

Based on our rigorous calculations, the value of the expression (cos 64° × cos 4° - cos 86° × cos 26°) ÷ (cos 71° × cos 3° + cos 87° × cos 19°) is 1.

Therefore, the correct answer among the given options is B. 1.

Kudos to everyone who followed along and hopefully managed to solve it. Keep practicing those identities, and you'll be a trig master in no time! Math is all about practice and understanding the underlying principles, and this problem was a fantastic way to reinforce those concepts. Keep exploring, keep questioning, and most importantly, keep enjoying the journey of mathematics!