Prove that tan^(-1)(1/7) + tan(1/13) = tan^(-1)(2/9)
Understand the Problem
The question is asking us to prove a trigonometric identity involving the tangent function. Specifically, it requires us to show that the sum of the arctangent of 1/7 and the tangent of 1/13 equals the arctangent of 2/9. The approach to solve this involves using the properties of inverse tangent and tangent addition formulas.
Answer
$$ \tan^{-1}\left(\frac{1}{7}\right) + \tan^{-1}\left(\frac{1}{13}\right) = \tan^{-1}\left(\frac{2}{9}\right) $$
Answer for screen readers
The identity is proven, showing that $$ \tan^{-1}\left(\frac{1}{7}\right) + \tan^{-1}\left(\frac{1}{13}\right) = \tan^{-1}\left(\frac{2}{9}\right) $$
Steps to Solve
- Use the Tangent Addition Formula
To prove the identity, we can use the tangent addition formula: $$ \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a) \tan(b)} $$
Let ( a = \tan^{-1}\left(\frac{1}{7}\right) ) and ( b = \tan^{-1}\left(\frac{1}{13}\right) ).
Calculating ( \tan(a) ) and ( \tan(b) ): $$ \tan(a) = \frac{1}{7} $$ $$ \tan(b) = \frac{1}{13} $$
- Substitute Values into the Formula
Now substitute these values into the tangent addition formula: $$ \tan(\tan^{-1}(\frac{1}{7}) + \tan^{-1}(\frac{1}{13})) = \frac{\frac{1}{7} + \frac{1}{13}}{1 - \frac{1}{7} \cdot \frac{1}{13}} $$
- Calculate the Numerator
Calculate the numerator: $$ \frac{1}{7} + \frac{1}{13} = \frac{13 + 7}{91} = \frac{20}{91} $$
- Calculate the Denominator
Calculate the denominator: $$ 1 - \frac{1}{7} \cdot \frac{1}{13} = 1 - \frac{1}{91} = \frac{91 - 1}{91} = \frac{90}{91} $$
- Combine Numerator and Denominator
Combine the results into the formula: $$ \tan(\tan^{-1}(\frac{1}{7}) + \tan^{-1}(\frac{1}{13})) = \frac{\frac{20}{91}}{\frac{90}{91}} = \frac{20}{90} = \frac{2}{9} $$
- Conclude the Proof
Since $$ \tan(\tan^{-1}(\frac{1}{7}) + \tan^{-1}(\frac{1}{13})) = \frac{2}{9} $$
It follows that: $$ \tan^{-1}\left(\frac{1}{7}\right) + \tan^{-1}\left(\frac{1}{13}\right) = \tan^{-1}\left(\frac{2}{9}\right) $$
Therefore, the identity is proven.
The identity is proven, showing that $$ \tan^{-1}\left(\frac{1}{7}\right) + \tan^{-1}\left(\frac{1}{13}\right) = \tan^{-1}\left(\frac{2}{9}\right) $$
More Information
This identity demonstrates the beauty of trigonometric relationships and how different angles can be combined through addition formulas. In this case, it shows that the sum of the arctangents of two fractions can be expressed as the arctangent of another fraction.
Tips
- Misapplying the Addition Formula: Some may forget to apply the tangent addition formula correctly. Always ensure that the formula is set up and manipulated appropriately.
- Arithmetic Errors: Mistakes during calculation of the numerator and denominator can lead to incorrect conclusions. Carefully verify each step.
- Ignoring Function Properties: Not recognizing that the functions involved are inverses can lead to confusion. Always remember the relationship between tangent and arctangent.
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