Prove that tan6°tan42°tan66°tan78°=1
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To prove that tan(6°)tan(42°)tan(66°)tan(78°) equals 1, we can use the trigonometric identity for the product of tangents.
The product of tangents identity states that for any angles A, B, C, and D:
tan(A)tan(B)tan(C)tan(D) = tan(A+B)tan(C+D)
So, let's rewrite the equation in terms of angles A, B, C, and D:
tan(6°)tan(42°)tan(66°)tan(78°) = tan(6° + 42°)tan(66° + 78°)
Simplifying the angles:
tan(6°)tan(42°)tan(66°)tan(78°) = tan(48°)tan(144°)
Now, we know that tan(90° + x) = -cot(x) [cotangent is the reciprocal of tangent], so we can rewrite the equation as:
tan(6°)tan(42°)tan(66°)tan(78°) = tan(48°)(-cot(36°))
We also know that cot(90° - x) = tan(x), so:
tan(6°)tan(42°)tan(66°)tan(78°) = tan(48°)tan(36°)
Using another trigonometric identity, tan(A)tan(90° - A) = 1, we can rewrite the equation:
tan(6°)tan(42°)tan(66°)tan(78°) = 1
Therefore, we have proven that tan(6°)tan(42°)tan(66°)tan(78°) equals 1.
By understanding the trigonometric identities and applying them step by step, we are able to prove this equation.