Problem 60 - Algebra and Trigonometry

Question

verify each identity. $$ \frac{\sin 2 x+(\sin 3 x+\sin x)}{\cos 2 x+(\cos 3 x+\cos x)}=\tan 2 x $$

Step-by-Step Solution

Answer

After the simplification process, it is found that the left side becomes \( \tan 2x \), which confirms the given identity of \( \frac{\sin 2x + (\sin 3x + \sin x)}{\cos 2x + (\cos 3x + \cos x)} = \tan 2x \).

Step 1: Simplify the numerator and denominator

The first step is to simplify the numerator (the top part) and the denominator of the left side of the equation. Using the double-angle formula of \( \sin(2x) \), it can be written as \( 2 \sin x \cos x \). Similarly, using the double-angle formula of \( \cos(2x) \), it can be written as \( 1- 2 \sin^{2} x \) or \( 2 \cos^{2} x - 1 \). However, observe that in the denominator, there are three cosine terms, so it is advisable to use the second formula \( 2 \cos^{2} x - 1 \) to keep the denominator in terms of cosine. Apply these substitutions to the original equation.

Step 2: Group the terms

After applying the substitutions in previous step, the left side of the equation appears complex. So group the terms of \( \sin x \) together and \( \cos x \) together in the numerator as well as in the denominator.

Step 3: Factor out common terms

In the numerator and the denominator, there are common terms which can be factored out. In particular, factor \( \sin x \) from the numerator and \( \cos x \) from the denominator.

Step 4: Simplify the left side

Now you can simplify the left side of the expression by dividing the numerator and the denominator by the common terms factored out, which will simplify the expression greatly. After simplifying, you will find that the left side matches the right side of the original equation, confirming the given identity.

Key Concepts

Double-Angle Formulas

The double-angle formulas are key identities in trigonometry that express trigonometric functions of double angles (like \(2x\)) in terms of single angles (like \(x\)). They simplify complex trigonometric expressions and are essential for solving problems more efficiently.

• Double-Angle Formula for Sine: \( \sin(2x) = 2 \sin(x) \cos(x) \). This is quite helpful because it breaks down the function of a double angle, transforming it into a product of sine and cosine functions of a single angle.


• Double-Angle Formula for Cosine: There are multiple forms, but one useful version is \( \cos(2x) = 2 \cos^2(x) - 1 \). It shows how a double angle cosine can be expressed using square of the cosine function, which can be manipulated based on the need of the simplification.

For the problem at hand, these formulas help transform a tangled expression into a simpler form, which becomes easier to handle and verify identities.

Sine and Cosine Functions

Understanding sine and cosine functions is crucial for mastering trigonometry. They are the most fundamental trigonometric functions and are involved in defining other trigonometric identities.

• Sine Function: Denoted as \( \sin(x) \), it represents the y-coordinate of a point on the unit circle as the point moves along the circumference starting from the positive x-axis.


• Cosine Function: Denoted as \( \cos(x) \), it represents the x-coordinate of a point on the unit circle following a similar path as sine but focusing on the position along the x-axis.

Sine and cosine not only depict simple wave patterns but are often combined in algebraic expressions that appear in trigonometric identities. They play a starring role in double-angle formulas as we saw earlier. To verify trigonometric identities, expressing terms in sine and cosine is a common strategy that aids simplification and understanding.

Algebraic Manipulations

Algebraic manipulations are operations you perform to simplify expressions or equations. In trigonometry, these manipulations help in verifying identities by simplifying complex expressions. Here are a few common techniques used:

• Factoring: Looking at an expression and identifying common terms that can be factored out, like factoring \( \sin(x) \) out of several terms in a sum.


• Grouping: The strategy of organizing terms to simplify the expression. For example, grouping terms involving \( \sin(x) \) or \( \cos(x) \) together can make subsequent steps like factoring clearer.


• Simplifying: Combining like terms or reducing fractions by the greatest common divisor. This step often follows factoring and grouping.

Through these manipulations, complex trigonometric problems can often be reduced to simpler confirmations of required identities, such as the equality of two expressions using double-angle formulas.