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Trigonometry Examples
Step 1
Rewrite the equation as .
Step 2
Step 2.1
Simplify .
Step 2.1.1
Simplify each term.
Step 2.1.1.1
Apply the sine double-angle identity.
Step 2.1.1.2
Cancel the common factor of .
Step 2.1.1.2.1
Cancel the common factor.
Step 2.1.1.2.2
Rewrite the expression.
Step 2.1.1.3
Cancel the common factor of .
Step 2.1.1.3.1
Cancel the common factor.
Step 2.1.1.3.2
Rewrite the expression.
Step 2.1.1.4
Convert from to .
Step 2.1.1.5
Convert from to .
Step 2.1.2
Add and .
Step 3
Step 3.1
Divide each term in by .
Step 3.2
Simplify the left side.
Step 3.2.1
Cancel the common factor of .
Step 3.2.1.1
Cancel the common factor.
Step 3.2.1.2
Divide by .
Step 4
Take the inverse secant of both sides of the equation to extract from inside the secant.
Step 5
Interchange the variables.
Step 6
Step 6.1
Rewrite the equation as .
Step 6.2
Take the inverse arcsecant of both sides of the equation to extract from inside the arcsecant.
Step 6.3
Multiply both sides of the equation by .
Step 6.4
Simplify the left side.
Step 6.4.1
Cancel the common factor of .
Step 6.4.1.1
Cancel the common factor.
Step 6.4.1.2
Rewrite the expression.
Step 7
Replace with to show the final answer.
Step 8
Step 8.1
To verify the inverse, check if and .
Step 8.2
Evaluate .
Step 8.2.1
Set up the composite result function.
Step 8.2.2
Evaluate by substituting in the value of into .
Step 8.2.3
The functions secant and arcsecant are inverses.
Step 8.2.4
Cancel the common factor of .
Step 8.2.4.1
Cancel the common factor.
Step 8.2.4.2
Rewrite the expression.
Step 8.3
Evaluate .
Step 8.3.1
Set up the composite result function.
Step 8.3.2
Evaluate by substituting in the value of into .
Step 8.3.3
Cancel the common factor of .
Step 8.3.3.1
Cancel the common factor.
Step 8.3.3.2
Divide by .
Step 8.4
Since and , then is the inverse of .