Given A + B + C = π, prove that A cot B + B cot A = cot C.

Understand the Problem

The question appears to be a mathematical problem involving trigonometric identities and possibly requires solving for a variable through a series of steps. The goal seems to be to establish relationships between angles A, B, and C based on cotangent functions.

Answer

The derived equation is $ A \cot B + B \cot A + \cot A \cot C = 1 $.

Steps to Solve

Set the Equations Start by noting the given equations:

( A + B + C = k )
( A + B = k - C )

Using Cotangent Identity Using the identity for cotangent, we have: $$ \cot(A + B) = \frac{\cot A \cot B - 1}{\cot A + \cot B} $$
Derive Further Relationships From the equations, set ( \cot(A + B) ): $$ \cot(A + B) = \cot(k - C) $$

This means we can also write: $$ \cot(A + B) = \cot(k - C) $$

Implement Relationships in Cotangent Using the cotangent identity, substitute: $$ \frac{\cot A \cot B - 1}{\cot A + \cot B} = -\cot C $$
Rearranging Terms Cross-multiply to eliminate the fraction: $$ \cot A \cot B - 1 = -\cot C (\cot A + \cot B) $$
Substituting Back This leads to: $$ \cot A \cot B + \cot C \cot A + \cot C \cot B = 1 $$
Solve for Cotangent Values Rearranging gives us the final cotangent relationship: $$ A \cot B + B \cot A + \cot A \cot C = 1 $$

The cotangent relationship derived is:

$$ A \cot B + B \cot A + \cot A \cot C = 1 $$

More Information

This solution highlights the interplay between angles and their cotangent functions in trigonometry. Such identities are useful in various mathematical problems, especially in geometry and calculus.

Tips

Misapplying Identities: Be cautious when using trigonometric identities. Always check the formulas to ensure proper application.
Neglecting Terms: In algebraic manipulation, it’s easy to miss terms. Carefully keep track of all components when rearranging equations.

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