Problem 87 - Precalculus Mathematics for Calculus

Question

Prove the identity. $$\frac{\sin 10 x}{\sin 9 x+\sin x}=\frac{\cos 5 x}{\cos 4 x}$$

Step-by-Step Solution

Answer

The identity is proven: both sides simplify to \(\frac{\cos 5x}{\cos 4x}\).

Step 1: Simplify the Left Side Using Sum-to-Product Formulas

Start by focusing on simplifying \(\sin 9x + \sin x\) which appears in the denominator on the left side of the identity. Use the sum-to-product formula for sine: \[\sin A + \sin B = 2 \sin \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right).\]Setting \(A = 9x\) and \(B = x\), we have:\[\sin 9x + \sin x = 2 \sin \left(\frac{10x}{2}\right) \cos \left(\frac{8x}{2}\right) = 2 \sin 5x \cos 4x.\]

Step 2: Substitute Back into the Left Side Expression

Substitute the result from Step 1 back into the left-hand side of the identity:\[\frac{\sin 10x}{\sin 9x + \sin x} = \frac{\sin 10x}{2 \sin 5x \cos 4x}.\]

Step 3: Simplify Using the Double Angle Formula

Use the double angle formula for sine: \[\sin 10x = 2 \sin 5x \cos 5x.\]Substitute this into the equation obtained in Step 2:\[\frac{2 \sin 5x \cos 5x}{2 \sin 5x \cos 4x} = \frac{\sin 5x \cos 5x}{\sin 5x \cos 4x}.\]

Step 4: Cancel Like Terms

In the expression from Step 3, \(\sin 5x\) in the numerator and denominator can be canceled:\[\frac{\cos 5x}{\cos 4x}.\]

Step 5: Compare Both Sides

The simplified left-hand side \(\frac{\cos 5x}{\cos 4x}\) matches exactly with the right-hand side of the original identity. This confirms the given identity is true.

Key Concepts

Sum-to-Product Formulas

Understanding sum-to-product formulas is crucial in simplifying trigonometric expressions where sums of sine or cosine functions are involved.
These formulas allow us to convert a sum of two trigonometric functions into a product, making expressions easier to manipulate and simplify.In our example, we began with the expression \( \sin 9x + \sin x \). Using the sum-to-product formula for sine, we transformed this into:

• \( \sin A + \sin B = 2 \sin \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right) \)

Here, \( A = 9x \) and \( B = x \). This transformation results in:

• \( 2 \sin 5x \cos 4x \)

This product form (\( 2 \sin 5x \cos 4x \)) is simpler to work with than the original sum, especially when dealing with identities or proving equations.

Double Angle Formulas

Double angle formulas play a vital role in simplifying trigonometric functions where angles are involved in double quantities such as \( 2x \), \( 3x \), or higher.
In this problem, we applied the double angle formula for sine to simplify the expression further.The double angle formula states:

• \( \sin 2\theta = 2 \sin \theta \cos \theta \)

In our example, we had \( \sin 10x \), which can be expressed using the double angle formula as:

• \( 2 \sin 5x \cos 5x \)

By substituting this into the fraction, we gained a more manageable form that allowed for further simplification steps. Double angle formulas are essential tools because they transform complex trigonometric expressions into forms that are easier to handle.

Simplification Techniques

Simplification techniques are key skills for solving trigonometric identities and equations.
They involve breaking down expressions into their simplest components.In this exercise, after applying sum-to-product and double angle formulas, the resulting expression was:

• \( \frac{2 \sin 5x \cos 5x}{2 \sin 5x \cos 4x} \)

We could then simplify by canceling out common terms.
In this case, \( \sin 5x \) in the numerator and denominator cancel each other out, which simplified the expression to:

• \( \frac{\cos 5x}{\cos 4x} \)

Simplification techniques also include factoring expressions, rationalizing, and using trigonometric identities effectively. The ultimate goal is to make the math manageable and reach a form where direct comparison with the desired result confirms its validity.