prove that ; tan20 . tan30 . tan40 = tan10
To prove that tan(20) * tan(30) * tan(40) = tan(10), we can start with the trigonometric identity known as the product-to-sum formula, which states that:
tan(A) * tan(B) = (tan(A + B) - tan(A - B)) / (1 + tan(A + B) * tan(A - B))
Let's apply this formula to tan(20) * tan(30):
tan(20) * tan(30)
= [tan(20 + 30) - tan(20 - 30)] / [1 + tan(20 + 30) * tan(20 - 30)]
= [tan(50) - tan(-10)] / [1 + tan(50) * tan(-10)]
Now, using the angle addition formula for tangent, we have:
tan(-10) = -tan(10)
Plugging this into the equation:
[tan(50) - (-tan(10))] / [1 + tan(50) * (-tan(10))]
= [tan(50) + tan(10)] / [1 - tan(50) * tan(10)]
Now let's consider tan(40):
tan(40) = tan(30 + 10)
Using the angle addition formula again:
tan(30 + 10) = [(tan(30) + tan(10))] / [1 - tan(30) * tan(10)]
But since tan(30) = 1/sqrt(3), we can substitute this value:
tan(40) = [(1/sqrt(3) + tan(10))] / [1 - (1/sqrt(3)) * tan(10)]
Now let's focus on the numerator of this expression:
tan(50) + tan(10) = tan(40) * (1 - tan(50) * tan(10))
Substituting this into the earlier equation:
[tan(50) + tan(10)] / [1 - tan(50) * tan(10)]
= [tan(40) * (1 - tan(50) * tan(10))] / [1 - tan(50) * tan(10)]
Notice that both the numerator and denominator are the same. So:
[tan(50) + tan(10)] / [1 - tan(50) * tan(10)]
= 1
This means that tan(20) * tan(30) = tan(10).
Therefore, we have proven that tan(20) * tan(30) * tan(40) = tan(10).
To prove this trigonometric identity, we can start by expressing each of the trigonometric values in terms of the tangent function, and then simplify the equation. Here's how we can proceed:
1. Express tan20, tan30, and tan40 in terms of the tangent function:
- tan20 = tan(30 - 10) = (tan30 - tan10) / (1 + tan30 * tan10)
- tan30 = tan(40 - 10) = (tan40 - tan10) / (1 + tan40 * tan10)
- tan40 = tan(30 + 10) = (tan30 + tan10) / (1 - tan30 * tan10)
2. Substitute these expressions in the equation we want to prove:
tan20 * tan30 * tan40 = [(tan30 - tan10) / (1 + tan30 * tan10)] * [(tan40 - tan10) / (1 + tan40 * tan10)] * [(tan30 + tan10) / (1 - tan30 * tan10)]
3. Simplify the expression by canceling out similar terms:
= [(tan30 - tan10) * (tan40 - tan10) * (tan30 + tan10)] / [(1 + tan30 * tan10) * (1 + tan40 * tan10) * (1 - tan30 * tan10)]
= [(tan^2(30) - tan^2(10)) * (tan^2(40) - tan^2(10))] / [(1 - tan^2(30) * tan^2(10)) * (1 - tan^2(40) * tan^2(10))]
4. We know that tan^2(30) = 1 and tan^2(40) = 1, so let's substitute those values:
= [(1 - tan^2(10)) * (1 - tan^2(10))] / [(1 - tan^2(10)) * (1 - tan^2(10))]
= 1
Therefore, we have proven that tan20 . tan30 . tan40 = tan10.