If `3sinx+4cosx=5`, then the value of `90tan^2(x/2)-60tan(x/2)+10` is equal to

Answer

To solve the equation \(3\sin x + 4\cos x = 5\) and find the value of \(90\tan^2\left(\frac{x}{2}\right) - 60\tan\left(\frac{x}{2}\right) + 10\), we will follow these steps:

### Step 1: Rewrite sine and cosine in terms of tangent
Using the formulas:
\[ \sin x = \frac{2\tan\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)} \]
\[ \cos x = \frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)} \]

### Step 2: Substitute into the equation
Substituting these into the equation:
\[ 3\left(\frac{2\tan\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)}\right) + 4\left(\frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)}\right) = 5 \]

### Step 3: Combine the terms
This simplifies to:
\[ \frac{6\tan\left(\frac{x}{2}\right) + 4 - 4\tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)} = 5 \]

### Step 4: Clear the denominator
Multiplying both sides by \(1 + \tan^2\left(\frac{x}{2}\right)\):
\[ 6\tan\left(\frac{x}{2}\right) + 4 - 4\tan^2\left(\frac{x}{2}\right) = 5 + 5\tan^2\left(\frac{x}{2}\right) \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ -4\tan^2\left(\frac{x}{2}\right) - 5\tan^2\left(\frac{x}{2}\right) + 6\tan\left(\frac{x}{2}\right) + 4 - 5 = 0 \]
\[ -9\tan^2\left(\frac{x}{2}\right) + 6\tan\left(\frac{x}{2}\right) - 1 = 0 \]

### Step 6: Multiply through by -1
To simplify, we multiply through by -1:
\[ 9\tan^2\left(\frac{x}{2}\right) - 6\tan\left(\frac{x}{2}\right) + 1 = 0 \]

### Step 7: Solve the quadratic equation
Let \(t = \tan\left(\frac{x}{2}\right)\):
\[ 9t^2 - 6t + 1 = 0 \]
Using the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 9 \cdot 1}}{2 \cdot 9} = \frac{6 \pm \sqrt{36 - 36}}{18} = \frac{6}{18} = \frac{1}{3} \]

### Step 8: Substitute back to find the value
Now substituting \(t = \frac{1}{3}\) back into the expression we need to evaluate:
\[ 90\tan^2\left(\frac{x}{2}\right) - 60\tan\left(\frac{x}{2}\right) + 10 \]
Substituting \(t\):
\[ 90\left(\frac{1}{3}\right)^2 - 60\left(\frac{1}{3}\right) + 10 = 90 \cdot \frac{1}{9} - 20 + 10 = 10 - 20 + 10 = 0 \]

### Final Answer
Thus, the value of \(90\tan^2\left(\frac{x}{2}\right) - 60\tan\left(\frac{x}{2}\right) + 10\) is:
\[ \boxed{0} \]