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Use double angle identity to simplify the following expression

tan 12 degrees/1-tan^2 12 degrees

User Ruakh
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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:

User Bhavin Chauhan
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