2tanx/(3+tan^x)=what
I'll assume you meant
2tanx/(3+tan^2x) = sin2x/(2+cos2x)
Not sure where you want to go with this. It can be massaged in many ways.
To solve the given equation 2tan(x)/(3+tan^2(x)), we will follow these steps:
Step 1: Simplify the equation by using the identity tan^2(x) = sec^2(x) - 1.
- The equation becomes: 2tan(x)/(3+(sec^2(x) - 1))
Step 2: Simplify further by using the identity sec^2(x) = 1 + tan^2(x).
- The equation becomes: 2tan(x)/(3+(1 + tan^2(x)) - 1)
Step 3: Simplify the denominator.
- The equation becomes: 2tan(x)/(3+1+tan^2(x) - 1)
= 2tan(x)/(4+tan^2(x))
Step 4: We can replace tan(x) with its reciprocal identity.
- Recall that tan(x) = sin(x)/cos(x).
- The equation becomes: 2(sin(x)/cos(x))/(4+(sin(x)/cos(x))^2)
Step 5: Simplify the equation and combine like terms.
- The equation becomes: (2sin(x))/(cos(x))(4+sin^2(x)/cos^2(x))
= 2sin(x)/(cos(x))(4cos^2(x) + sin^2(x))
Step 6: Simplify further by using the identity sin^2(x) + cos^2(x) = 1.
- The equation becomes: 2sin(x)/(cos(x))(4cos^2(x) + (1 - cos^2(x)))
= 2sin(x)/(cos(x))(3cos^2(x) + 1)
Step 7: Simplify by canceling common factors.
- The equation becomes: 2sin(x)/(3cos^3(x) + cos(x))
So, the simplified form of the given equation 2tanx/(3+tan^x) is 2sin(x)/(3cos^3(x) + cos(x)).