Trigonometry Examples

Solve for x csc(x)-cot(2x)=tan(x)
Step 1
Simplify the left side.
Tap for more steps...
Step 1.1
Simplify each term.
Tap for more steps...
Step 1.1.1
Rewrite in terms of sines and cosines.
Step 1.1.2
Rewrite in terms of sines and cosines.
Step 2
Simplify the right side.
Tap for more steps...
Step 2.1
Rewrite in terms of sines and cosines.
Step 3
Multiply both sides of the equation by .
Step 4
Apply the distributive property.
Step 5
Cancel the common factor of .
Tap for more steps...
Step 5.1
Cancel the common factor.
Step 5.2
Rewrite the expression.
Step 6
Rewrite using the commutative property of multiplication.
Step 7
Simplify each term.
Tap for more steps...
Step 7.1
Combine and .
Step 7.2
Use the double-angle identity to transform to .
Step 7.3
Apply the sine double-angle identity.
Step 7.4
Cancel the common factor of .
Tap for more steps...
Step 7.4.1
Cancel the common factor.
Step 7.4.2
Rewrite the expression.
Step 7.5
Apply the cosine double-angle identity.
Step 8
Multiply .
Tap for more steps...
Step 8.1
Combine and .
Step 8.2
Raise to the power of .
Step 8.3
Raise to the power of .
Step 8.4
Use the power rule to combine exponents.
Step 8.5
Add and .
Step 9
Subtract from both sides of the equation.
Step 10
Simplify each term.
Tap for more steps...
Step 10.1
Separate fractions.
Step 10.2
Rewrite as a product.
Step 10.3
Write as a fraction with denominator .
Step 10.4
Simplify.
Tap for more steps...
Step 10.4.1
Divide by .
Step 10.4.2
Convert from to .
Step 10.5
Multiply .
Tap for more steps...
Step 10.5.1
Combine and .
Step 10.5.2
Combine and .
Step 10.6
Factor out of .
Step 10.7
Separate fractions.
Step 10.8
Convert from to .
Step 10.9
Divide by .
Step 11
Simplify the left side.
Tap for more steps...
Step 11.1
Simplify each term.
Tap for more steps...
Step 11.1.1
Rewrite in terms of sines and cosines.
Step 11.1.2
Combine and .
Step 11.1.3
Multiply the numerator by the reciprocal of the denominator.
Step 11.1.4
Multiply by .
Step 11.1.5
Move to the left of .
Step 11.1.6
Rewrite in terms of sines and cosines.
Step 11.1.7
Multiply .
Tap for more steps...
Step 11.1.7.1
Combine and .
Step 11.1.7.2
Raise to the power of .
Step 11.1.7.3
Raise to the power of .
Step 11.1.7.4
Use the power rule to combine exponents.
Step 11.1.7.5
Add and .
Step 12
Multiply both sides of the equation by .
Step 13
Apply the distributive property.
Step 14
Simplify.
Tap for more steps...
Step 14.1
Multiply by .
Step 14.2
Rewrite using the commutative property of multiplication.
Step 14.3
Rewrite using the commutative property of multiplication.
Step 15
Simplify each term.
Tap for more steps...
Step 15.1
Cancel the common factor of .
Tap for more steps...
Step 15.1.1
Factor out of .
Step 15.1.2
Factor out of .
Step 15.1.3
Cancel the common factor.
Step 15.1.4
Rewrite the expression.
Step 15.2
Cancel the common factor of .
Tap for more steps...
Step 15.2.1
Factor out of .
Step 15.2.2
Cancel the common factor.
Step 15.2.3
Rewrite the expression.
Step 16
Multiply by .
Step 17
Replace with .
Step 18
Use the double-angle identity to transform to .
Step 19
Simplify the left side.
Tap for more steps...
Step 19.1
Simplify .
Tap for more steps...
Step 19.1.1
Apply pythagorean identity.
Step 19.1.2
To write as a fraction with a common denominator, multiply by .
Step 19.1.3
Simplify terms.
Tap for more steps...
Step 19.1.3.1
Combine and .
Step 19.1.3.2
Combine the numerators over the common denominator.
Step 19.1.4
Simplify the numerator.
Tap for more steps...
Step 19.1.4.1
Move to the left of .
Step 19.1.4.2
Apply the cosine double-angle identity.
Step 19.1.4.3
Simplify each term.
Tap for more steps...
Step 19.1.4.3.1
Use the double-angle identity to transform to .
Step 19.1.4.3.2
Apply the distributive property.
Step 19.1.4.3.3
Multiply by .
Step 19.1.4.3.4
Multiply by .
Step 19.1.5
To write as a fraction with a common denominator, multiply by .
Step 19.1.6
Simplify terms.
Tap for more steps...
Step 19.1.6.1
Combine and .
Step 19.1.6.2
Combine the numerators over the common denominator.
Step 19.1.7
Simplify the numerator.
Tap for more steps...
Step 19.1.7.1
Multiply by .
Step 19.1.7.2
Move .
Step 19.1.7.3
Factor out of .
Step 19.1.7.4
Factor out of .
Step 19.1.7.5
Factor out of .
Step 19.1.7.6
Rearrange terms.
Step 19.1.7.7
Apply pythagorean identity.
Step 19.1.7.8
Multiply by .
Step 19.1.7.9
Add and .
Step 20
Solve the equation for .
Tap for more steps...
Step 20.1
Set the numerator equal to zero.
Step 20.2
Solve the equation for .
Tap for more steps...
Step 20.2.1
Add to both sides of the equation.
Step 20.2.2
Divide each term in by and simplify.
Tap for more steps...
Step 20.2.2.1
Divide each term in by .
Step 20.2.2.2
Simplify the left side.
Tap for more steps...
Step 20.2.2.2.1
Cancel the common factor of .
Tap for more steps...
Step 20.2.2.2.1.1
Cancel the common factor.
Step 20.2.2.2.1.2
Divide by .
Step 20.2.3
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 20.2.4
Simplify the right side.
Tap for more steps...
Step 20.2.4.1
The exact value of is .
Step 20.2.5
The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the fourth quadrant.
Step 20.2.6
Simplify .
Tap for more steps...
Step 20.2.6.1
To write as a fraction with a common denominator, multiply by .
Step 20.2.6.2
Combine fractions.
Tap for more steps...
Step 20.2.6.2.1
Combine and .
Step 20.2.6.2.2
Combine the numerators over the common denominator.
Step 20.2.6.3
Simplify the numerator.
Tap for more steps...
Step 20.2.6.3.1
Multiply by .
Step 20.2.6.3.2
Subtract from .
Step 20.2.7
Find the period of .
Tap for more steps...
Step 20.2.7.1
The period of the function can be calculated using .
Step 20.2.7.2
Replace with in the formula for period.
Step 20.2.7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 20.2.7.4
Divide by .
Step 20.2.8
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
, for any integer