Problem 42 - Contemporary Precalculus

Question

State whether or not the equation is an identity. If it is an identity, prove it. $$\frac{\cos (-x)}{\sin (-x)}=-\cot x$$

Step-by-Step Solution

Answer

Answer: Yes, the given equation is an identity as we have shown that both sides are equal for all values of x for which both sides are defined.

Step 1: Rewrite the left-hand side using the properties of cosine and sine of a negative angle

Recall the properties of cosine and sine for negative angles: $$\cos(-x) = \cos(x)$$ $$\sin(-x) = -\sin(x)$$ Now we can rewrite the left-hand side of the equation using these properties: $$\frac{\cos (-x)}{\sin (-x)} = \frac{\cos (x)}{-\sin (x)}$$

Step 2: Rewrite the right-hand side as a fraction using the definition of cotangent

The cotangent function can be defined as the reciprocal of the tangent function. Therefore, we can rewrite the right-hand side of the equation in terms of sine and cosine: $$\cot x = \frac{\cos x}{\sin x}$$ By our step 1, we have $$\frac{\cos(x)}{-\sin(x)}$$ and for step 2, we have $$-\frac{\cos(x)}{\sin(x)}$$ which means that both sides of the equation are equal. Now, let's go back to our initial equation: $$\frac{\cos (-x)}{\sin (-x)}=-\cot x$$ We have shown that the left-hand side and the right-hand side are equal: $$\frac{\cos (x)}{-\sin (x)} = - \frac{\cos(x)}{\sin(x)}$$ So, the given equation is an identity as both sides are equal for all values of x for which both sides are defined.

Key Concepts

Cosine of Negative Angle

Understanding the behavior of the cosine function when dealing with negative angles is pivotal in trigonometry. The key property to remember is that cosine is an even function. This means that for any angle x, the value of cosine remains the same even if the angle is made negative:
\[ \cos(-x) = \cos(x) \].
What's particularly useful about this property is its simplifying power in trigonometric equations and identities. When faced with a negative angle within a cosine function, you can easily replace it with the positive counterpart without changing the value of the expression. This property directly influences the symmetry of the cosine graph on a coordinate plane, as it mirrors perfectly along the y-axis. This symmetry is part of what makes the unit circle approach to trigonometry so elegant and powerful.

Sine of Negative Angle

The sine function, although similar to cosine in many regards, behaves differently with negative angles. Specifically, sine is an odd function, resulting in the sine of a negative angle being the negative of the sine of the positive angle:
\[ \sin(-x) = -\sin(x) \].
This property is significant when simplifying expressions or proving identities, as it makes dealing with negative angles straightforward—simply factor out the negative sign and proceed with the positive angle. Just like cosine's property shapes its graph's symmetry, the property of sine also determines its graph's symmetry about the origin—the graph of sine is invariant under a rotation of 180 degrees or \( \pi \) radians. This reflection in the unit circle is why the concept of sine is so intertwined with circular and oscillatory motion.

Cotangent Function

Diving into the cotangent function, we enter the realm of reciprocal trigonometric functions. Cotangent is the reciprocal of the tangent function, which itself is the ratio of sine to cosine. The definition of cotangent can be written as:
\[ \cot x = \frac{\cos x}{\sin x} \].
The geometrical significance of cotangent lies in its representation of the ratio of the adjacent side to the opposite side in a right-angled triangle, but only when considering angles different from 0, \( \pi/2 \) (or 90 degrees), and \( \pi \) (or 180 degrees). When simplifying trigonometric expressions or solving equations, recognizing that cotangent equates to the quotient of cosine over sine can be crucial. It also means that multiplying both sides of a cotangent function by negative one affects both the numerator and denominator identically, resulting in the original positive cotangent function being negated. This concept plays a fundamental role when proving trigonometric identities, such as the one where we showed that \( \frac{\cos (-x)}{\sin (-x)} = -\cot x \) is indeed a true identity.