How to Differentiate 'tan^{-1}(1) - cos(x)/sin(x)' with Respect to 'x'

In this article, we will explore the process of differentiating the expression y tan^{-1}(1) - cos(x)/sin(x) with respect to x. This step-by-step guide will help you understand the underlying trigonometric identities and calculus principles involved in this problem.

Introduction to the Expression

The given expression can be broken down into two parts for simplification. Let's start by identifying the components of the expression:

tan^{-1}(1): This term represents the inverse tangent of 1, which is equal to (frac{pi}{4}). cos(x)/sin(x): This can be simplified to cot(x), the cotangent of x.

Therefore, the expression y tan^{-1}(1) - cos(x)/sin(x) simplifies to:

y (frac{pi}{4}) - cot(x)

Step-by-Step Differentiation

Now, let's differentiate the simplified expression with respect to x.

Identify the constants and apply the rules of differentiation:

The derivative of a constant is zero, so the term (frac{pi}{4}) will disappear when differentiated.

Differentiate the cotangent term:

The derivative of cot(x) is known to be -csc^2(x).

Combine the results:

The expression now simplifies to:

dy/dx -csc^2(x)

Final Result

Therefore, the derivative of the given expression y tan^{-1}(1) - cos(x)/sin(x) with respect to x is:

dy/dx -csc^2(x)

Advanced Techniques and Variations

For a more in-depth understanding, let's walk through an advanced approach using trigonometric identities and algebraic simplification.

Consider the expression:

tan^{-1}left(frac{1}{cos(x)/sin(x)}right)

This can be rewritten using trigonometric identities:

(tan^{-1}left(frac{1}{cot(x)}right) tan^{-1}(tan(frac{pi}{2} - x)) frac{pi}{2} - x)

The derivative of (frac{pi}{2} - x) is:

(frac{d}{dx}left(frac{pi}{2} - xright) -frac{1}{2})

Therefore, the differentiated expression is:

(frac{d}{dx}left(frac{1}{cos(x)/sin(x)}right) -frac{1}{2})

Conclusion

Differentiating complex trigonometric expressions like tan^{-1}(1) - cos(x)/sin(x) requires a solid understanding of trigonometric identities and the rules of differentiation. The final result is:

dy/dx -csc^2(x)

Key Takeaways

Trigonometric identities can significantly simplify expressions. The derivative of a constant is zero. The derivative of cot(x) is -csc^2(x).

FAQ

Q: What is the importance of understanding trigonometric differentiation?

A: Understanding trigonometric differentiation is crucial for solving complex calculus problems in various fields such as physics, engineering, and mathematics. It helps in modeling periodic phenomena and analyzing wave-like behaviors.

Q: Can this differentiation process be applied to other inverse trigonometric functions?

A: Yes, similar techniques can be applied to other inverse trigonometric functions, such as arcsin(x) and arctan(x), by using their respective identities and differentiation rules.