Exploring the Trigonometric Expression: cos 75° cos 15° / (cos 75° - cos 15°)

Understanding and solving trigonometric expressions is a fundamental component in the field of mathematics and has extensive applications in various scientific and engineering disciplines. One such expression that showcases the elegance and complexity of trigonometric identities is:

cos 75° cos 15° / (cos 75° - cos 15°)

Solving the Expression Using Cosine Addition and Subtraction Formulas

The expression can be evaluated using the cosine addition and subtraction formulas:

Cosine Addition and Subtraction Formulas

The cosine addition formula is:

cos A cos B 2 cos(AB/2) cos(A-B/2)

The cosine subtraction formula is:

cos A - cos B -2 sin(AB/2) sin(A-B/2)

Step 1: Calculate and Substitute A and B

Let's set A 75° and B 15°.

Step 2: Apply the Formulas

Now, applying the cosine formulas:

Using the cosine addition formula:

cos 75° cos 15° 2 cos(90°/2) cos(60°/2) 2 cos 45° cos 30°

Using the known values:

cos 45° sqrt{2}/2 and cos 30° sqrt{3}/2

2 * (sqrt{2}/2) * (sqrt{3}/2) sqrt{6}/2

Using the cosine subtraction formula:

cos 75° - cos 15° -2 sin(90°/2) sin(60°/2) -2 sin 45° sin 30°

Using the known values:

sin 45° sqrt{2}/2 and sin 30° 1/2

-2 * (sqrt{2}/2) * (1/2) -sqrt{2}/2

Step 3: Substitute Back into the Original Expression

Substituting these values back into the original expression:

(cos 75° cos 15°) / (cos 75° - cos 15°) (sqrt{6}/2) / (-sqrt{2}/2) sqrt{6}/(-sqrt{2}) -sqrt{3}

Final Answer

Therefore, the value of cos 75° cos 15° / (cos 75° - cos 15°) is:

-sqrt{3}

Note: An alternative approach to solving this expression involves using half-angle formulas for 30°. Another method involves rationalizing the numerator, demonstrating the versatility of trigonometric identities.

Understanding and solving such expressions not only enhances one's mathematical skills but also provides insights into the underlying symmetry and patterns in trigonometry.