Proving the Trigonometric Identity: sin x sin 3x sin 5x sin 7x 4 cos x cos 2x sin 4x

Introduction

In this article, we will explore and provide a detailed proof of the trigonometric identity:

sin x sin 3x sin 5x sin 7x 4 cos x cos 2x sin 4x

This identity is not only interesting but also demonstrates the power of trigonometric identities and their applications. Let's dive into the proof step-by-step.

Proof of the Identity

To prove the identity, we will start with the left-hand side (LHS) and show that it simplifies to the right-hand side (RHS).

Step 1: Grouping Terms

We can group the sine terms into pairs:

sin x sin 7x sin 3x sin 5x

Step 2: Applying Sum-to-Product Identities

Using the sum-to-product identities:

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

For sin x sin 7x:

A x B 7x

Therefore:

sin x sin 7x 2 sin(8x)/2 cos(-6x)/2 2 sin 4x cos 3x

For sin 3x sin 5x:

A 3x B 5x

Therefore:

sin 3x sin 5x 2 sin(8x)/2 cos(-2x)/2 2 sin 4x cos x

Step 3: Combine the Results

Now, we can combine the results from both pairs:

sin x sin 7x sin 3x sin 5x (2 sin 4x cos 3x)(2 sin 4x cos x)

Factoring out 2 sin 4x, we get:

2 sin 4x (2 cos 3x cos x)

Step 4: Simplifying cos 3x cos x

We can apply the sum-to-product identity again to simplify cos 3x cos x:

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

For cos 3x cos x:

A 3x B x

Therefore:

cos 3x cos x 2 cos(4x)/2 cos(2x)/2 2 cos 2x cos x

Step 5: Substitute Back

Substituting this back into our expression, we have:

2 sin 4x (2 cos 2x cos x) 4 cos 2x cos x sin 4x

Conclusion

Thus, we have shown that:

sin x sin 3x sin 5x sin 7x 4 cos x cos 2x sin 4x

This completes the proof of the given trigonometric identity. The journey through this proof highlights the elegance and utility of trigonometric identities in simplifying complex expressions.

Additional Resources

For further exploration, you can refer to the following resources:

Proofs and Applications of Sum-to-Product Identities Trigonometry Practice Problems Advanced Trigonometric Identities

Understanding these identities can help in solving a variety of mathematical problems and deepen your knowledge in trigonometry.