Question

Verify each potential identity by graphing, and then prove the identity. a) \(\frac{\csc x}{2 \cos x}=\csc 2 x\) b) \(\frac{\sin x \cos x}{1+\cos x}=\frac{1-\cos x}{\tan x}\) c) \(\frac{\sin x+\tan x}{1+\cos x}=\frac{\sin 2 x}{2 \cos ^{2} x}\)

Short answer

Graph the identities, convert to sine and cosine, simplify, and prove algebraically.

Step-by-step solution

Graph Each Side of the Identity

To verify if the identities are correct, graph both sides of each identity on the same set of axes using graphing software or a graphing calculator. If the graphs overlap, then the identity is likely true.

Convert to Basic Trigonometric Functions

Rewrite each side of the identities using basic trigonometric functions (sine and cosine).

Simplify the Expressions

Simplify both sides of each identity, looking for common terms that can be simplified or eliminated.

Verify Algebraically

Show that both sides of the identity are equal by algebraic manipulations and trigonometric identities.

Key concepts

Trigonometric Functions

Understanding trigonometric functions is essential in verifying identities. Trigonometric functions relate the angles of a triangle to the lengths of its sides. The most important functions are sine (\(\text{sin}\)), cosine (\(\text{cos}\)), tangent (\(\text{tan}\)), cotangent (\(\text{cot}\)), secant (\(\text{sec}\)), and cosecant (\(\text{csc}\)).
Each function has a unique relationship with the others, which allows us to convert one into another and simplify expressions. For instance, we know that \(\text{csc} x\) is the reciprocal of \(\text{sin} x\), and \(\text{sec} x\) is the reciprocal of \(\text{cos} x\). These relationships are key in solving and verifying trigonometric identities.
In our example problems, identities a), b), and c) involve these trigonometric functions. To verify and simplify them, we rely heavily on these basic relationships.

Verifying Identities

Verifying trigonometric identities involves confirming that two different expressions are, in fact, equal for all values of the variable where both sides are defined. There are many strategies to verify identities, but one of the first steps is usually graphical verification.
In the given exercise, the first step is to graph both sides of each identity on the same coordinate system. If the graphs overlap perfectly across the range of values, it strongly suggests that the identity is true. This graphical approach gives a visual confirmation, though it's not as rigorous as algebraic proof.
After visually confirming, the next step is to algebraically manipulate the expressions to prove the identities. This requires detailed knowledge of trigonometric identities and properties. For example, recognizing and applying \(\text{Pythagorean identities}\) can help simplify expressions and reveal their equivalence.

Graphical Verification

Graphical verification is an effective preliminary step in verifying trigonometric identities. By plotting the graphs of both sides of the identity, we can verify if they match.
For example, to verify \(\frac{\text{csc} x}{2 \text{cos} x} = \text{csc} 2x\) graphically, we need to:
1. Plot \(\frac{\text{csc} x}{2 \text{cos} x}\) on a graph.
2. Plot \(\text{csc} 2x\) on the same graph.
If the two graphs completely overlap, then the identity holds true over the plotted range.
Graphing is not a foolproof method as small errors or restrictions in graphing software might mislead, but it's a valuable initial check before formal algebraic proof.

Algebraic Manipulation

Algebraic manipulation is crucial in proving trigonometric identities. This involves rewriting expressions using known identities and reducing them to show equivalence.
Here's a brief outline of the process involved in the given exercise:
1. **Convert to Basic Functions**: Rewrite the given trigonometric expressions in terms of sine and cosine. This simplifies the manipulation.
For instance: \(\frac{\text{csc} x}{2 \text{cos} x} = \frac{1 / \text{sin} x}{2 \text{cos} x} \).
2. **Simplify Expressions**: Simplify both sides separately, using trigonometric identities like \(\text{cos}^2 x + \text{sin}^2 x = 1\).
3. **Eliminate and Reduce**: Look for ways to cancel terms or combine similar elements.
4. **Verify Equality**: Show that both simplified sides are identical.
This combination of steps helps in proving the verity of each trigonometric identity beyond the graphical verification.