given that tan(x/2)=a. Express tan(x) and sin(x) in terms of a. Hint: use that fact that x= 2*(x/2)
To express tan(x) and sin(x) in terms of a, we can use the double-angle formula for tangent:
tan(2θ) = (2tanθ) / (1 - tan²θ)
Using the hint given, we can substitute θ = x/2 into the formula to find the value of tan(x). Let's do the calculation step by step:
1. Start with the given equation: tan(x/2) = a
2. Multiply both sides of the equation by 1 - tan²(x/2):
(1 - tan²(x/2)) * tan(x/2) = a * (1 - tan²(x/2))
3. Simplify the left side using the identity: 1 - tan²(x/2) = sec²(x/2):
sec²(x/2) * tan(x/2) = a * sec²(x/2)
4. Divide both sides of the equation by sec²(x/2) to isolate tan(x/2):
tan(x/2) = (a * sec²(x/2)) / sec²(x/2)
tan(x/2) = a
5. Substitute the value of tan(x/2) obtained from step 4 into the double-angle formula:
tan(x) = 2 * tan(x/2) / (1 - tan²(x/2))
tan(x) = 2 * a / (1 - a²)
Now, let's proceed to express sin(x) in terms of a. We can use the identity:
sin(x) = 2 * sin(x/2) * cos(x/2)
Using the half-angle formula for sine, we can substitute the value of tan(x/2) = a into the formula:
1. Start with the half-angle formula: sin(x) = 2 * sin(x/2) * cos(x/2)
2. Substitute sin(x/2) = (2 * tan(x/2)) / (1 + tan²(x/2)) and cos(x/2) = 1 / √(1 + tan²(x/2)):
sin(x) = 2 * (2 * a / (1 + a²)) * (1 / √(1 + a²))
sin(x) = 4a / (√(1 + a²) * (1 + a²))
Hence, tan(x) = 2a / (1 - a²) and sin(x) = 4a / (√(1 + a²) * (1 + a²)), expressed in terms of a.