Proving the Identity: cos^2 45° - sin^2 15° √3/4

In this article, we will explore and prove the trigonometric identity that cos^2 45° - sin^2 15° √3/4. This process will involve breaking down the problem into several steps, utilizing fundamental trigonometric principles and properties to derive the given identity.

Introduction to Trigonometric Identities

Trigonometric identities are fundamental relationships between trigonometric functions that are always true. These identities are crucial in simplifying and solving complex trigonometric problems. One such identity we will be working with in this article is the difference of squares identity, which is a powerful tool in verifying such expressions.

Step-by-Step Solution

To demonstrate that cos^2 45° - sin^2 15° √3/4, we need to break down the expression into simpler components and use known trigonometric values. Let's proceed step by step.

Step 1: Calculate cos^2 45°

The cosine of 45° is a well-known value in trigonometry:

cos 45° (frac{sqrt{2}}{2})

To find cos^2 45°, we square this value:

cos^2 45° (left(frac{sqrt{2}}{2}right)^2 frac{2}{4} frac{1}{2})

Step 2: Calculate sin 15°

For sin 15°, we use the sine subtraction formula:

Sin A - B sin A cos B - cos A sin B

Here, A 45° and B 30°.

sin 15° sin 45° cos 30° - cos 45° sin 30°

Substituting the values:

sin 45° (frac{sqrt{2}}{2})

cos 30° (frac{sqrt{3}}{2})

cos 45° (frac{sqrt{2}}{2})

sin 30° (frac{1}{2})

sin 15° (frac{sqrt{2}}{2} cdot frac{sqrt{3}}{2} - frac{sqrt{2}}{2} cdot frac{1}{2})

sin 15° (frac{sqrt{6}}{4} - frac{sqrt{2}}{4})

sin 15° (frac{sqrt{6} - sqrt{2}}{4})

Step 3: Calculate sin^2 15°

Now we square sin 15°:

sin^2 15° (left(frac{sqrt{6} - sqrt{2}}{4}right)^2)

sin^2 15° (frac{6 - 2sqrt{12} 2}{16})

sin^2 15° (frac{8 - 4sqrt{3}}{16} frac{2 - sqrt{3}}{4})

Step 4: Calculate cos^2 45° - sin^2 15°

Now, substitute the values we calculated:

cos^2 45° - sin^2 15° (frac{1}{2} - frac{2 - sqrt{3}}{4})

Express (frac{1}{2}) with a denominator of 4:

(frac{1}{2} frac{2}{4})

Combine the fractions:

(frac{2}{4} - frac{2 - sqrt{3}}{4} frac{2 - 2 sqrt{3}}{4} frac{sqrt{3}}{4})

Conclusion

We have successfully demonstrated that cos^2 45° - sin^2 15° √3/4, verifying the given identity.

Additional Insights

This problem showcases the importance of trigonometric identities in solving complex expressions. Familiarity with these identities, such as the difference of squares and sine subtraction formulas, is crucial for simplifying and verifying trigonometric expressions.

Understanding and applying these identities not only strengthens your mathematical skills but also aids in tackling more advanced problems in calculus, physics, and engineering.

By practicing similar problems, you can enhance your proficiency in using trigonometric identities and build a robust foundation for more advanced concepts in mathematics.