Solve the equation for cos(3theta/2).tan(3theta/2)
To solve the equation for cos(3theta/2) * tan(3theta/2), we first need to determine the values of cos(3theta/2) and tan(3theta/2) separately.
Let's first find the value of cos(3theta/2):
1. Recall that cos(2theta) can be expressed using the double-angle identity as cos(2theta) = 2cos^2(theta) - 1.
2. Similarly, we can express cos(3theta) using the triple-angle identity as cos(3theta) = 4cos^3(theta) - 3cos(theta).
3. By substituting x = theta/2, we can rewrite cos(3theta) as cos(3x) = 4cos^3(x) - 3cos(x).
4. Now, let's find cos(3x/2). We can express cos(3x) as cos(3x) = cos((3x/2) + (3x/2)) using the sum-angle identity.
5. Applying the sum-angle identity, cos(3x) = cos((3x/2) + (3x/2)) = cos(3x/2)cos(3x/2) - sin(3x/2)sin(3x/2).
6. Simplifying further, cos(3x) = cos^2(3x/2) - sin^2(3x/2).
Next, let's find the value of tan(3theta/2):
1. Recall that tangent is defined as tan(x) = sin(x) / cos(x).
2. Thus, tan(3x/2) = sin(3x/2) / cos(3x/2).
Now that we have expressions for cos(3theta/2) and tan(3theta/2), we substitute them back into the equation cos(3theta/2) * tan(3theta/2), giving us (cos^2(3x/2) - sin^2(3x/2)) * (sin(3x/2) / cos(3x/2)).
To further simplify this equation, we can divide both the numerator and denominator by cos(3x/2):
= (cos^2(3x/2) - sin^2(3x/2)) * (sin(3x/2) / cos(3x/2))
= (cos^2(3x/2) - sin^2(3x/2)) * sin(3x/2) / (cos(3x/2) / cos(3x/2))
= (cos^2(3x/2) - sin^2(3x/2)) * sin(3x/2) / 1
= (cos^2(3x/2) - sin^2(3x/2)) * sin(3x/2)
= cos^2(3x/2) * sin(3x/2) - sin^3(3x/2).
So, the equation for cos(3theta/2) * tan(3theta/2) is cos^2(3x/2) * sin(3x/2) - sin^3(3x/2).