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Trigonometry Examples
Step 1
Interchange the variables.
Step 2
Step 2.1
Rewrite the equation as .
Step 2.2
Add to both sides of the equation.
Step 2.3
Take the inverse cosecant of both sides of the equation to extract from inside the cosecant.
Step 2.4
Divide each term in by and simplify.
Step 2.4.1
Divide each term in by .
Step 2.4.2
Simplify the left side.
Step 2.4.2.1
Cancel the common factor of .
Step 2.4.2.1.1
Cancel the common factor.
Step 2.4.2.1.2
Divide by .
Step 3
Replace with to show the final answer.
Step 4
Step 4.1
To verify the inverse, check if and .
Step 4.2
Evaluate .
Step 4.2.1
Set up the composite result function.
Step 4.2.2
Evaluate by substituting in the value of into .
Step 4.2.3
Combine the opposite terms in .
Step 4.2.3.1
Add and .
Step 4.2.3.2
Add and .
Step 4.3
Evaluate .
Step 4.3.1
Set up the composite result function.
Step 4.3.2
Evaluate by substituting in the value of into .
Step 4.3.3
Simplify each term.
Step 4.3.3.1
Cancel the common factor of .
Step 4.3.3.1.1
Cancel the common factor.
Step 4.3.3.1.2
Rewrite the expression.
Step 4.3.3.2
The functions cosecant and arccosecant are inverses.
Step 4.3.4
Combine the opposite terms in .
Step 4.3.4.1
Subtract from .
Step 4.3.4.2
Add and .
Step 4.4
Since and , then is the inverse of .