simplify the expression by using a double angle formula or a half angle formula:
a. 2 tan 7 degrees /1-tan^2 7 degrees
b.2 tan 7 theta / 1-tan^2 7 theta
recall that tan(2x) = 2tanx/(1-tan^2 x)
To simplify the expression in part a, we can utilize the double angle formula for tangent:
tan(2θ) = 2tan(θ) / (1 - tan^2(θ))
Comparing this formula to the expression given:
2tan(7 degrees) / (1 - tan^2(7 degrees))
We can see that the expression matches the double angle formula, with θ being equal to 7 degrees. Therefore, we can substitute θ = 7 degrees into the formula to simplify the expression:
2tan(7 degrees) / (1 - tan^2(7 degrees)) = tan(2 * 7 degrees) = tan(14 degrees)
So, the expression simplifies to tan(14 degrees).
Now let's simplify the expression in part b. Similar to part a, we can use the double angle formula for tangent:
tan(2θ) = 2tan(θ) / (1 - tan^2(θ))
Comparing it to the given expression:
2tan(7θ) / (1 - tan^2(7θ))
We can substitute θ = 7θ into the formula to simplify the expression:
2tan(7θ) / (1 - tan^2(7θ)) = tan(2 * 7θ) = tan(14θ)
So, the expression simplifies to tan(14θ).