If tan(1/2 x)=b express tanx in terms of b
From tan (2A) = 2tanA/(1 - tan^2 A) we get
tanx = 2tan(1/2 x)/(1 - tan^2 (1/2 x))
= 2b/(1-b^2)
tan(1/2 x) = b
We know that
tan(A + B) = (tanA + tanB)/(1 - tanAtanB)
so
tanx = tan(1/2 x + 1/2 x) =
[tan(1/2 x) + tan(1/2 x) ] / 1 - ( tan(1/2 x)tan(1/2 x) )
= (b + b) / 1 - (bxb)
= 2b / (1 - b²)
Well, let's take a humorous approach to this problem, shall we?
Alright, now let's think about this. We have tan(1/2 x) = b, and we want to express tanx in terms of b.
So, to do that, let's imagine we're at a party. You walk up to tanx and say, "Hey, tanx, my friend b has been asking about you. What's up?" And tanx looks at you and replies, "Oh, b! Yeah, I remember that guy. Well, here's the deal - I can express myself in terms of b."
Confused, you ask, "Alright, tanx, spill the beans! How can you do that?"
Grinning, tanx says, "No worries, my friend. Remember that tan(1/2 x) = b? Well, it turns out that I can express myself as 2b / (1 - b^2)."
You blink in surprise, processing this information. "Wait, tanx, is this for real?"
With a mischievous smirk, tanx nods and says, "Absolutely! Just try it out for yourself and you'll see."
And there you have it! tanx = 2b / (1 - b^2). Just remember, this answer comes with a side order of humor!
To express tan(x) in terms of b, we can use the double angle formula for tangent:
tan(2θ) = (2tanθ) / (1 - tan²θ).
Let's replace θ with 1/2 x:
tan(x) = tan(2(1/2 x)) = (2tan(1/2 x)) / (1 - tan²(1/2 x)).
Since we are given that tan(1/2 x) = b, we can substitute that in:
tan(x) = (2b) / (1 - b²).
Therefore, tan(x) is expressed in terms of b as (2b) / (1 - b²).
To express tan(x) in terms of b, we can use the half-angle identity for tangent. The half-angle identity states that:
tan(1/2 x) = sqrt((1 - cos(x)) / (1 + cos(x)))
From the given equation, tan(1/2 x) = b, we can substitute b into the equation:
b = sqrt((1 - cos(x)) / (1 + cos(x)))
Now, we can solve for cos(x):
b^2 = (1 - cos(x)) / (1 + cos(x))
To simplify this expression, we can multiply both sides of the equation by (1 + cos(x)):
b^2(1 + cos(x)) = 1 - cos(x)
Expanding the equation, we have:
b^2 + b^2*cos(x) = 1 - cos(x)
Moving all the terms involving cos(x) to one side:
b^2 + cos(x) * (b^2 + 1) = 1
Now, we can isolate cos(x) by subtracting b^2 and 1 from both sides:
cos(x) = (1 - b^2) / (b^2 + 1)
Finally, we can use the identity tan(x) = sin(x) / cos(x) to express tan(x) in terms of b:
tan(x) = sin(x) / cos(x)
= sin(x) / ((1 - b^2) / (b^2 + 1))
Further simplification is possible depending on the context of the problem.