Problem 101 - Algebra and Trigonometry

Question

Show that the equation is not an identity. $$ \begin{array}{l}{\text { (a) } \sin 2 x=2 \sin x} & {\text { (b) } \sin (x+y)=\sin x+\sin y} \\ {\text { (c) } \sec ^{2} x+\csc ^{2} x=1} \\ {\text { (d) } \frac{1}{\sin x+\cos x}=\csc x+\sec x}\end{array} $$

Step-by-Step Solution

Answer

None of the equations are identities. They fail at specific values.

Step 1: Understanding Identities

The problem asks us to determine if certain equations are identities. An identity is an equation that holds true for all values of the variable(s) involved. To show that an equation is not an identity, we need to find at least one instance where the equation does not hold true.

Step 2: Checking Equation (a): \( \sin 2x = 2 \sin x \)

Recall the double-angle identity for sine: \( \sin 2x = 2 \sin x \cos x \). Comparing this with \( 2 \sin x \), it's evident that \( \sin 2x = 2 \sin x \cos x eq 2 \sin x \) for all \( x \). For example, try \( x = \frac{\pi}{4} \): \( \sin 2(\frac{\pi}{4}) = \sqrt{2}/2 \), while \( 2 \sin(\frac{\pi}{4}) = \sqrt{2} \), showing the equation fails.

Step 3: Checking Equation (b): \( \sin(x+y) = \sin x + \sin y \)

The addition formula for sine states that \( \sin(x+y) = \sin x \cos y + \cos x \sin y \). This differs from \( \sin x + \sin y \). A counterexample would be \( x = y = \frac{\pi}{4} \): \( \sin(\frac{\pi}{2}) = 1 \), but \( \sin(\frac{\pi}{4}) + \sin(\frac{\pi}{4}) = \sqrt{2} \), thus proving the equation is false.

Step 4: Checking Equation (c): \( \sec^2 x + \csc^2 x = 1 \)

Recollect the definitions \( \sec x = \frac{1}{\cos x} \) and \( \csc x = \frac{1}{\sin x} \). Hence, \( \sec^2 x = \frac{1}{\cos^2 x} \) and \( \csc^2 x = \frac{1}{\sin^2 x} \), such that \( \sec^2 x + \csc^2 x = \frac{1}{\cos^2 x} + \frac{1}{\sin^2 x} eq 1 \) for all \( x \). For \( x = \frac{\pi}{4} \), \( \sec^2(\frac{\pi}{4}) + \csc^2(\frac{\pi}{4}) = 4 eq 1 \).

Step 5: Checking Equation (d): \( \frac{1}{\sin x + \cos x} = \csc x + \sec x \)

First, find a value where each side should be simplified differently. When \( x = \frac{\pi}{4} \), the left-hand side equals \( \frac{1}{\sin(\frac{\pi}{4}) + \cos(\frac{\pi}{4})} = \frac{1}{\sqrt{2}} \), which is \( \frac{1}{\sqrt{2} + \sqrt{2}} = \frac{1}{\sqrt{2}} \), and the right-hand side equals \( 2\sqrt{2} \) (since \( \csc(\frac{\pi}{4}) = \sec(\frac{\pi}{4}) = \sqrt{2} \)). Clearly, the two sides are not equal, proving the equation is not an identity.

Key Concepts

Double Angle Formulas

Double angle formulas are essential tools in trigonometry, aimed at expressing trigonometric functions of double angles in terms of single angles. The expressions for these identities can help in simplifying and solving trigonometric equations.
For instance, the double angle formula for sine is given by:

• \( \sin 2x = 2 \sin x \cos x \)

This identity tells you that the sine of double an angle \(2x\) can be expressed as twice the product of the sine and cosine of \(x\).
When trying to determine if an equation is an identity, applying these formulas can quickly reveal inconsistencies. Consider the false identity \(\sin 2x = 2 \sin x\). Here, if you apply the double angle formula, it is clear that for an equation to be valid, \( \cos x \) must be equal to 1 for all \( x \), which is impossible since \( \cos x \) varies between -1 and 1.
Understanding these formulas is crucial to mastering trigonometry and analyzing how multiple trigonometric functions relate to each other in complex trigonometric expressions.

Trigonometric Functions

Trigonometric functions are fundamental components of mathematics, especially in studying angles and their relationships. The primary trigonometric functions include sine (\( \sin \)), cosine (\( \cos \)), tangent (\( \tan \)), as well as their reciprocals: cosecant (\( \csc \)), secant (\( \sec \)), and cotangent (\( \cot \)).
These functions relate the angles of a triangle to the lengths of its sides. For example:

• \( \sin x = \frac{\text{opposite}}{\text{hypotenuse}} \)
• \( \cos x = \frac{\text{adjacent}}{\text{hypotenuse}} \)

Reciprocal functions:

• \( \csc x = \frac{1}{\sin x} \)
• \( \sec x = \frac{1}{\cos x} \)

When applied incorrectly, such as with the equation \( \sec^2 x + \csc^2 x = 1 \), highlighting misunderstandings in reciprocal relationships can make apparent why an equation is not an identity. When such values are evaluated correctly for specific angles, like \( x = \frac{\pi}{4} \), the disparity between expected and actual values rub out the possibility of them being true identities.
Knowing these functions inside out allows one to solve a wide array of mathematical problems, from simple triangles to wave motion analysis.

Angle Sum Identities

Angle sum identities are powerful equations used to find the trigonometric functions of compound angles. They express the sine, cosine, and tangent of an angle that is the sum or difference of two angles in terms of the trigonometric functions of each angle.
Take, for instance, the angle sum identity for sine:

• \( \sin(x+y) = \sin x \cos y + \cos x \sin y \)

This formula can debunk an equation like \( \sin(x+y) = \sin x + \sin y \), which assumes a simple addition of terms. To demonstrate the invalidity of this equation, use a numerical example. If both angles are \( \frac{\pi}{4} \), the sum \(\sin(\frac{\pi}{2}) = 1\) differs from \(2\sin(\frac{\pi}{4}) = \sqrt{2} \).
Angle sum identities not only offer insight into the intricacies of trigonometric relationships but also serve as foundational tools in proving or disproving identities. Knowing when and how to apply these identities effectively can significantly enhance problem-solving skills in mathematics.