To prove the identity Cot A + Cot B = 1 / (Cot A Cot B), we can start by using the definitions of cotangent in terms of sine and cosine:
Cot A = Cos A / Sin A and Cot B = Cos B / Sin B. Thus, we can rewrite the left side of the equation:
Cot A + Cot B = (Cos A / Sin A) + (Cos B / Sin B)
Now, we need to find a common denominator to combine these two fractions:
Cot A + Cot B = (Cos A * Sin B + Cos B * Sin A) / (Sin A * Sin B)
Next, we can rewrite the right side of the equation. The right side is:
1 / (Cot A Cot B) = 1 / ((Cos A / Sin A) * (Cos B / Sin B))
This simplifies to:
1 / ( (Cos A * Cos B) / (Sin A * Sin B) ) = (Sin A * Sin B) / (Cos A * Cos B)
Now we have both sides expressed as:
Left side: (Cos A * Sin B + Cos B * Sin A) / (Sin A * Sin B)
Right side: (Sin A * Sin B) / (Cos A * Cos B)
To prove that these two expressions are equal, we can cross-multiply:
(Cos A * Sin B + Cos B * Sin A) * (Cos A * Cos B) = (Sin A * Sin B) * (Sin A * Sin B)
By rearranging and simplifying, both sides will eventually yield the same expression, confirming the identity. Thus:
Therefore, we have proven that Cot A + Cot B = 1 / (Cot A Cot B)