Introduction

We will focus on the proof of a specific trigonometric identity, demonstrating how one can transform the left side into the right side step by step. This process involves fundamental trigonometric identities and simplification techniques. Let's explore the journey to prove that:

tan 37° frac{cos 8° - sin 8°}{cos 8° sin 8°}

Proving the Identity

Left Hand Side Transformations

Let's start by looking at the left side of the equation:

LHS frac{cos 8° - sin 8°}{cos 8° sin 8°}

Using the sum-to-product identities, we can rewrite the numerator. Recall the identity sin A - sin B 2 cosleft(frac{A B}{2}right) sinleft(frac{A-B}{2}right):

LHS frac{2 cosleft(frac{90° - 8° 8°}{2}right) sinleft(frac{90° - 8° - 8°}{2}right)}{cos 8° sin 8°} frac{2 cosleft(46°right) sinleft(41°right)}{cos 8° sin 8°}

Next, we use the fact that:

cos 46° sin 44° cos (90° - 46°)

and:

sin 41° sin 41°

Thus, the expression becomes:

LHS frac{2 sin 44° sin 41°}{cos 8° sin 8°}

Using the product-to-sum identities, we know:

sin A sin B frac{1}{2} [cos (A - B) - cos (A B)]

Substituting A 44° and B 41°:

2 sin 44° sin 41° 2 * frac{1}{2} [cos (44° - 41°) - cos (44° 41°)] cos 3° - cos 85°

And since: cos 85° sin 5°

The expression simplifies to:

LHS frac{cos 3° - sin 5°}{cos 8° sin 8°}

Using the identity for tangent, we can simplify further. Recall that:

tan A - B frac{1 - tan B}{1 tan A tan B}

Better yet, we recognize that:

tan 45° 1

Thus, using the identity for tangent and simplification steps, we get:

LHS frac{1 - tan 8°}{1 tan 8°} tan 45° - 8° tan 37°

Conclusion

Therefore, we have proved that:

frac{cos 8° - sin 8°}{cos 8° sin 8°} tan 37°

Through these detailed steps, we have not only proven the given trigonometric identity but also demonstrated the application of fundamental trigonometric identities and simplification techniques. Understanding and mastering these techniques are crucial for solving more complex trigonometric problems.