Answer: We can use the double angle identity for tangent, which states that:
tan(2θ) = 2tan(θ) / (1 - tan²(θ))
to simplify the expression.
Let θ = 6 degrees, then we have:
tan(2θ) = tan(12 degrees)
tan(2θ) = 2tan(θ) / (1 - tan²(θ))
tan(12 degrees) = 2tan(6 degrees) / (1 - tan²(6 degrees))
We can use the tangent half-angle identity to find tan(6 degrees), which states that:
tan(θ/2) = sin(θ) / (1 + cos(θ))
Letting θ = 12 degrees, we get:
tan(6 degrees) = sin(12 degrees) / (1 + cos(12 degrees))
We can then use the double angle identity for sine, which states that:
sin(2θ) = 2sin(θ)cos(θ)
to simplify sin(12 degrees). Letting θ = 6 degrees, we get:
sin(12 degrees) = 2sin(6 degrees)cos(6 degrees)
We can use the half-angle identity for cosine to find cos(6 degrees), which states that:
cos(θ/2) = √((1 + cos(θ)) / 2)
Letting θ = 12 degrees, we get:
cos(6 degrees) = √((1 + cos(12 degrees)) / 2)
Substituting these values into the original expression, we get:
tan(12 degrees) = 2tan(6 degrees) / (1 - tan²(6 degrees))
tan(12 degrees) = 2(sin(12 degrees) / (1 + cos(12 degrees))) / (1 - (sin²(12 degrees) / (1 + cos(12 degrees))²))
tan(12 degrees) = 2(2sin(6 degrees)cos(6 degrees) / (1 + cos(12 degrees))) / (1 - (4sin²(6 degrees)cos²(6 degrees) / (1 + cos(12 degrees))²))
tan(12 degrees) = (4sin(6 degrees)cos(6 degrees)) / (1 + cos(12 degrees) - 4sin²(6 degrees)cos²(6 degrees))
This is the simplified expression using the double angle identity.
Explanation: