The angle addition and subtraction formulas for tangent are invaluable tools in trigonometry. These formulas allow us to express \( \tan(A \pm B) \) in terms of \( \tan(A) \) and \( \tan(B) \). In this article, we will derive these formulas using the angle addition and subtraction formulas for sine and cosine.
Angle Addition and Subtraction Formulas for Sine and Cosine
Before we proceed, let’s recall the angle addition and subtraction formulas for sine and cosine:
\[
\sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B)
\]
\[
\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)
\]
The angle addition formula for tangent is:
\[
\tan(A + B) = \frac{\sin(A + B)}{\cos(A + B)}
\]
Derivation
Using the angle addition formulas for sine and cosine, we get:
\[
\tan(A + B) = \frac{\sin(A)\cos(B) + \cos(A)\sin(B)}{\cos(A)\cos(B) – \sin(A)\sin(B)}
\]
Now, we divide both the numerator and the denominator by \( \cos(A)\cos(B) \):
\[
\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 – \tan(A)\tan(B)}
\]
Deriving the Angle Subtraction Formula for Tangent
The angle subtraction formula for tangent can also be derived using the angle addition and subtraction formulas for sine and cosine. The formula is:
\[
\tan(A – B) = \frac{\tan A – \tan B}{1 + \tan A \tan B}
\]
Deriving the Double Angle Formula for Tangent
The double angle formula for tangent can be derived using the angle addition formula for tangent, which is:
\[
\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 – \tan(A)\tan(B)}
\]
\[
\tan(\theta + \theta) = \frac{\tan(\theta) + \tan(\theta)}{1 – \tan(\theta)\tan(\theta)}
\]
\[
\tan(2\theta) = \frac{2\tan \theta}{1 – \tan^2 \theta}
\]
Conclusion
We have successfully derived the angle addition formula for tangent, paying special attention to the step where we divide by \( \cos(A)\cos(B) \). This formula is essential for solving a wide range of trigonometric problems and has applications in various fields such as physics, engineering, and computer science.
