Question

Use a trigonometric identity to show that the derivatives of the inverse cotangent and inverse cosecant differ from the derivatives of the inverse tangent and inverse secant, respectively, by a multiplicative factor of -1

Short answer

How do the derivatives of the inverse cosecant and inverse secant functions differ? Answer: The derivatives of the inverse cotangent and inverse tangent functions differ by a multiplicative factor of -1. Similarly, the derivatives of the inverse cosecant and inverse secant functions also differ by a multiplicative factor of -1.

Step-by-step solution

Differentiate inverse cotangent function.

To differentiate the inverse cotangent function, we can apply the differentiation formula for inverse trigonometric functions, as follows: Let y = arccot(x), then x = cot(y). Thus, \\[ \frac{dx}{dy} = -\csc^2(y) \Rightarrow \frac{dy}{dx} = -\frac{1}{\csc^2(y)} \\] Using the Pythagorean identity for cotangent, we get \\[ \cot^2(y) + 1 = \csc^2(y) \Rightarrow \csc^2(y) = 1 + \cot^2(y) = 1 + x^2\\] So the derivative of the inverse cotangent will be, \\[ \frac{dy}{dx} = -\frac{1}{1 + x^2} \\]

Differentiate inverse tangent function.

To differentiate the inverse tangent function, similar to step one, let y = arctan(x), then x = tan(y). Using the chain rule and Pythagorean identity for tangent, \\[ \frac{dx}{dy} = \sec^2(y) \Rightarrow \frac{dy}{dx} = \frac{1}{\sec^2(y)} \\] \\[ \sec^2(y) = 1 + \tan^2(y) = 1 + x^2 \\] \\[ \frac{dy}{dx} = \frac{1}{1 + x^2} \\]

Differentiate inverse cosecant function.

To differentiate the inverse cosecant function, let y = arccsc(x), then x = csc(y). Differentiating and using the Pythagorean identity for cosecant, we get: \\[ \frac{dx}{dy} = -\csc(y)\cot(y) \Rightarrow \frac{dy}{dx} = -\frac{1}{\csc(y)\cot(y)} \\] \\[ \csc^2(y) - 1 = \cot^2(y) \Rightarrow \cot(y) = \sqrt{\csc^2(y) - 1} = \sqrt{x^2 - 1} \\] \\[ \frac{dy}{dx} = -\frac{1}{x\sqrt{x^2 - 1}} \\]

Differentiate inverse secant function.

To differentiate the inverse secant function, let y = arcsec(x), then x = sec(y). Differentiating and using the Pythagorean identity for secant, we get: \\[ \frac{dx}{dy} = \sec(y)\tan(y) \Rightarrow \frac{dy}{dx} = \frac{1}{\sec(y)\tan(y)} \\] \\[ \tan^2(y) + 1 = \sec^2(y) \Rightarrow \tan(y) = \sqrt{\sec^2(y) - 1} = \sqrt{x^2 - 1} \\] \\[ \frac{dy}{dx} = \frac{1}{x\sqrt{x^2 - 1}} \\]

Compare the derivatives.

Comparing the derivatives of the inverse cotangent and the inverse tangent functions, we have: \\[ \frac{d(\text{arccot}(x))}{dx} = -\frac{1}{1 + x^2} \\] and \\[ \frac{d(\text{arctan}(x))}{dx} = \frac{1}{1 + x^2} \\] Comparing the derivatives of the inverse cosecant and inverse secant functions, we have: \\[ \frac{d(\text{arccsc}(x))}{dx} = -\frac{1}{x\sqrt{x^2 - 1}} \\] and \\[ \frac{d(\text{arcsec}(x))}{dx} = \frac{1}{x\sqrt{x^2 - 1}} \\]

Conclusion

From the comparisons, we can conclude that the derivatives of the inverse cotangent and inverse cosecant functions differ from the derivatives of the inverse tangent and inverse secant functions, respectively, by a multiplicative factor of -1.

Key concepts

Inverse Cotangent Derivative

Understanding the derivative of the inverse cotangent function involves a few steps. First, let's symbolize inverse cotangent as arccot(x). The task is to find the rate of change of this angle as the value of x changes, which mathematically is represented as \(\frac{d}{dx}arccot(x)\).

Using the relationship between the cotangent and its reciprocal function, cosecant squared, we use the Pythagorean identity to express cosecant squared in terms of cotangent squared.

\[ \csc^2(y) = 1 + \cot^2(y) \]
Now, substituting x for cot(y), we get \(\csc^2(y) = 1 + x^2\). This expression allows us to easily find the derivative, which ultimately turns out to be:

\[ \frac{d}{dx}arccot(x) = -\frac{1}{1 + x^2} \]
The negative sign indicates that as x increases, the angle y decreases, which is characteristic of the cotangent function as it is a decreasing function.

Inverse Cosecant Derivative

Now, let's delve into the inverse cosecant function derivative. Representing the inverse cosecant as arccsc(x), the goal is to find \(\frac{d}{dx}arccsc(x)\).

We start the differentiation by considering the relationship x = csc(y). With differentiation, we arrive at an expression involving cosecant and cotangent, and then, using the Pythagorean identify, we solve for cotangent in terms of cosecant.

\[ \cot(y) = \sqrt{\csc^2(y) - 1} = \sqrt{x^2 - 1} \]
With these steps, the derivative of the inverse cosecant simplifies to:

\[ \frac{d}{dx}arccsc(x) = -\frac{1}{x\sqrt{x^2 - 1}} \]
Like the inverse cotangent, this derivative carries a negative sign as well, which is consistent with the decreasing nature of the cosecant function.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. These identities are powerful tools used to simplify and evaluate trigonometric expressions, solve equations, and calculate derivatives.

Some of the most commonly used trigonometric identities include the reciprocal identities, ratio identities, and Pythagorean identities. Reciprocal identities relate trigonometric functions to their reciprocals, for example, \(\sin(\theta) = \frac{1}{\csc(\theta)}\). Ratio identities, such as \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\), show the relationship between different functions.

When dealing with derivatives of inverse trigonometric functions, Pythagorean identities are particularly vital as they allow us to express one trigonometric function in terms of another, thus facilitating the differentiation process.

Pythagorean Identity

The Pythagorean identity is one of the cornerstones in trigonometry and is derived from the Pythagorean theorem. It states that for any angle \(\theta\), the square of the sine plus the square of the cosine equals one:

\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
This identity can be manipulated to develop other identities, such as those used in the differentiation of inverse trigonometric functions. For example, by dividing through by \(\cos^2(\theta)\), we obtain the identity \(\tan^2(\theta) + 1 = \sec^2(\theta)\), and similarly for cotangent and cosecant. These identities are vital when converting from one function to another, enabling us to simplify complex expressions and calculate derivatives of inverse trigonometric functions.