Sin x inverse tan x + cos x inverse cot x
Understand the Problem
The question is asking to find the value of the expression 'Sinx inverse tanx + cosx inverse cotx'. This expression involves trigonometric functions and their inverses, suggesting a need to simplify or evaluate it using properties of trigonometric identities.
Answer
$\frac{\pi}{2}$
Answer for screen readers
The value of the expression is $\frac{\pi}{2}$.
Steps to Solve
- Identify the expression
The given expression is $\arcsin(\tan x) + \arccos(\cot x)$.
- Simplify the components
The $\tan x$ function can be expressed in terms of sine and cosine: $$ \tan x = \frac{\sin x}{\cos x} $$
For $\cot x$, we can rewrite it as: $$ \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} $$
- Rewrite the expression using these identities
Substituting the definitions in the expression: $$ \arcsin\left(\frac{\sin x}{\cos x}\right) + \arccos\left(\frac{\cos x}{\sin x}\right) $$
- Analyze the arcsin and arccos functions
We know from trigonometric identities:
- $\arcsin(y) + \arccos(y) = \frac{\pi}{2}$ for $y \in [-1, 1]$
- Observe the ranges
Both $\tan x$ and $\cot x$ are defined and have values within valid ranges when $x$ lies in appropriate quadrants. We can find a common value $y$ to express the arcsin and arccos functions.
- Simplify to find the final result
We apply the identity: If $y = \tan x$, then $\arcsin(\tan x) + \arccos(\cot x) = \frac{\pi}{2}$ under the condition that $x$ is appropriate for the values to fall within valid ranges.
The value of the expression is $\frac{\pi}{2}$.
More Information
The identity used to find the final value shows a fundamental relationship between inverse trigonometric functions. This identity is useful in many areas of mathematics, particularly in calculus and analysis.
Tips
- Confusing the domains of the functions: Ensure that when evaluating inverse trigonometric functions, the input values must be within the proper range for the output to be valid.
- Not recognizing the relationship $\arcsin(y) + \arccos(y) = \frac{\pi}{2}$, which could lead to incomplete solutions.
