Problem 101 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

None of the given equations are identities as they do not hold for all values.

Step 1: Verify Identity (a)

To check if \(\sin 2x = 2\sin x\) is an identity, we use the double angle formula for sine: \(\sin 2x = 2\sin x \cos x\). Comparing both sides, \(2\sin x\) and \(2\sin x \cos x\), it is clear since \(\cos x\) is not always 1, the equation is not an identity.

Step 2: Verify Identity (b)

For \(\sin(x+y) = \sin x + \sin y\), we use the angle addition formula: \(\sin(x+y) = \sin x \cos y + \cos x \sin y\). The equation \(\sin x + \sin y\) simplifies to \(\sin x \cdot 1 + 1 \cdot \sin y\), so generally, both sides are equal only when \(\cos y = 1\) and \(\cos x = 1\). Therefore, this is not an identity.

Step 3: Verify Identity (c)

Evaluate \(\sec^2 x + \csc^2 x = 1\). We have: \(\sec^2 x = \frac{1}{\cos^2 x}\) and \(\csc^2 x = \frac{1}{\sin^2 x}\). Consider the simplest case \(x = \frac{\pi}{4}\), then \(\sec^2 x + \csc^2 x\) equals \(2\), not \(1\), proof the equation is not an identity.

Step 4: Verify Identity (d)

For \(\frac{1}{\sin x + \cos x} = \csc x + \sec x\), note \(\csc x = \frac{1}{\sin x}\) and \(\sec x = \frac{1}{\cos x}\). For verification, try \(x = \frac{\pi}{4}\), giving \(\frac{1}{\sqrt{2}} = \sqrt{2}\). The sides are not equal, thus this equation is not an identity.

Key Concepts

Double Angle Formula

The double angle formula is a powerful trigonometric tool that simplifies expressions involving trigonometric functions of double angles. The formula for sine is expressed as \[ \sin 2x = 2 \sin x \cos x \].
This indicates that the sine of double an angle is equal to double the product of sine and cosine of half the angle. This formula helps transform and simplify complex trigonometric expressions.

• Example: To express \( \sin 60° \) as a double angle, we consider \( \sin 30° = \frac{1}{2} \) and \( \cos 30° = \frac{\sqrt{3}}{2} \). Therefore, \( \sin 60° = 2 \times \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \).

Understanding this formula highlights why the expression \( \sin 2x = 2 \sin x \) is not an identity. The presence of \( \cos x \) in the double angle formula means this factor cannot be ignored or assumed to be 1 for all values of \( x \). Thus, the equation fails for most angles less than or greater than 90°.

Angle Addition Formula

The angle addition formula allows the calculation of the sine or cosine of the sum of two angles. For sine, it is expressed as \[ \sin(x+y) = \sin x \cos y + \cos x \sin y \].
This formula helps to simplify expressions by breaking them into components that are easier to handle.

• Illustration: If \( x = 30° \) and \( y = 45° \), \( \sin(75°) \) can be calculated as \( \sin(30° + 45°) = \sin 30° \cos 45° + \cos 30° \sin 45° \), resulting in the sum \( \frac{1}{2} \times \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2} \), simplifying to \( \frac{\sqrt{6} + \sqrt{2}}{4} \).

In comparison to \( \sin(x+y) = \sin x + \sin y \), it is clear that the direct addition of the sines misses these extra terms - \( \cos x \sin y \) and \( \sin x \cos y \). This explains why the expression is not universally true.

Secant Function

The secant function is the reciprocal of the cosine function, defined as \[ \sec x = \frac{1}{\cos x} \].
This identity is useful in simplifying expressions that include cosine, particularly in certain trigonometric identities and equations.

• For \( x = 0° \), \( \sec x \) equals 1 since \( \cos x = 1 \).
• For \( x = 60° \), \( \cos x = \frac{1}{2}\), thus \( \sec x = 2 \).

Realizing the importance of secant, we understand its role in illustrating why the expression \(\sec^2 x + \csc^2 x = 1 \) fails. Since both secant and cosecant are greater or equal to 1, their combined squared values exceed 1, unlike the given equation. When checking for specific angles, like \( x = \frac{\pi}{4} \), the outcome challenges the validity of such an identity.

Cosecant Function

Cosecant is the reciprocal of the sine function, expressed as \[ \csc x = \frac{1}{\sin x} \].
It plays a significant role when sine values are low, giving it large outcomes.

• When \( x = 30° \), \( \sin x = \frac{1}{2} \), hence \( \csc x = 2 \).
• At \( x = 90° \), \( \sin x = 1 \), leading to \( \csc x = 1 \).

This brings clarity to inequalities and identities involving sine. When looking at the given equation \( \frac{1}{\sin x + \cos x} = \csc x + \sec x \), the collective values of \( \csc x \) and \( \sec x \) are noticeably high for most angles, contrasting sharply with the left-hand side under typical conditions like \( x = \frac{\pi}{4} \), thus it doesn't satisfy the form of an identity.