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The formula \(\tan A + \tan B = \frac{\sin(A+B)}{\cos A \cdot \cos B}\) is a trigonometric identity that expresses the sum of tangents of two angles in terms of sine and cosine functions of those angles.
The calculator uses the trigonometric identity:
Where:
Explanation: This identity is derived from the fundamental definitions of trigonometric functions and is useful for simplifying trigonometric expressions involving sums of tangents.
Details: Trigonometric identities like this one are essential tools in mathematics, physics, and engineering for simplifying complex expressions, solving equations, and proving other mathematical theorems.
Tips: Enter values for sin(A+B), cos A, and cos B. All values must be between -1 and 1 (inclusive), and cos A and cos B cannot be zero as division by zero is undefined.
Q1: Why can't cos A or cos B be zero?
A: When cos A or cos B equals zero, the denominator becomes zero, making the expression undefined as division by zero is not allowed in mathematics.
Q2: What are the valid ranges for input values?
A: Sine and cosine functions output values between -1 and 1, so input values should be within this range.
Q3: When is this identity particularly useful?
A: This identity is useful when simplifying trigonometric expressions, solving trigonometric equations, or when working with problems involving sums of tangent functions.
Q4: Can this formula be derived from basic trigonometric definitions?
A: Yes, starting from \(\tan A = \frac{\sin A}{\cos A}\) and \(\tan B = \frac{\sin B}{\cos B}\), and using the sine addition formula \(\sin(A+B) = \sin A \cos B + \cos A \sin B\).
Q5: Are there similar identities for other trigonometric functions?
A: Yes, there are numerous trigonometric identities for sums and differences of various trigonometric functions, such as sum-to-product and product-to-sum formulas.