Trigonometry Examples

Solve for x (1-tan(x)^2)/(1+tan(x)^2)+1=2cos(x)^2
Step 1
Simplify the left side.
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Step 1.1
Simplify each term.
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Step 1.1.1
Rearrange terms.
Step 1.1.2
Apply pythagorean identity.
Step 1.1.3
Simplify the numerator.
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Step 1.1.3.1
Rewrite as .
Step 1.1.3.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 2
Move all terms containing to the left side of the equation.
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Step 2.1
Subtract from both sides of the equation.
Step 2.2
Simplify each term.
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Step 2.2.1
Simplify the numerator.
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Step 2.2.1.1
Rewrite in terms of sines and cosines.
Step 2.2.1.2
Rewrite in terms of sines and cosines.
Step 2.2.2
Simplify the denominator.
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Step 2.2.2.1
Rewrite in terms of sines and cosines.
Step 2.2.2.2
Apply the product rule to .
Step 2.2.2.3
One to any power is one.
Step 2.2.3
Multiply the numerator by the reciprocal of the denominator.
Step 2.2.4
Expand using the FOIL Method.
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Step 2.2.4.1
Apply the distributive property.
Step 2.2.4.2
Apply the distributive property.
Step 2.2.4.3
Apply the distributive property.
Step 2.2.5
Simplify and combine like terms.
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Step 2.2.5.1
Simplify each term.
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Step 2.2.5.1.1
Multiply by .
Step 2.2.5.1.2
Multiply by .
Step 2.2.5.1.3
Multiply by .
Step 2.2.5.1.4
Rewrite using the commutative property of multiplication.
Step 2.2.5.1.5
Multiply .
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Step 2.2.5.1.5.1
Multiply by .
Step 2.2.5.1.5.2
Raise to the power of .
Step 2.2.5.1.5.3
Raise to the power of .
Step 2.2.5.1.5.4
Use the power rule to combine exponents.
Step 2.2.5.1.5.5
Add and .
Step 2.2.5.1.5.6
Raise to the power of .
Step 2.2.5.1.5.7
Raise to the power of .
Step 2.2.5.1.5.8
Use the power rule to combine exponents.
Step 2.2.5.1.5.9
Add and .
Step 2.2.5.2
Add and .
Step 2.2.5.3
Add and .
Step 2.2.6
Apply the distributive property.
Step 2.2.7
Multiply by .
Step 2.2.8
Cancel the common factor of .
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Step 2.2.8.1
Move the leading negative in into the numerator.
Step 2.2.8.2
Cancel the common factor.
Step 2.2.8.3
Rewrite the expression.
Step 2.2.9
Apply the cosine double-angle identity.
Step 3
Use the double-angle identity to transform to .
Step 4
Move all terms not containing to the right side of the equation.
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Step 4.1
Subtract from both sides of the equation.
Step 4.2
Subtract from both sides of the equation.
Step 5
Simplify the left side.
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Step 5.1
Simplify .
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Step 5.1.1
Simplify with factoring out.
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Step 5.1.1.1
Factor out of .
Step 5.1.1.2
Factor out of .
Step 5.1.1.3
Factor out of .
Step 5.1.2
Apply pythagorean identity.
Step 5.1.3
Multiply by .
Step 6
Simplify the right side.
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Step 6.1
Subtract from .
Step 7
Since , the equation will always be true for any value of .
All real numbers
Step 8
The result can be shown in multiple forms.
All real numbers
Interval Notation: