To address the given problem, we will verify the equation step by step.
Let's consider the equation:
[tex]\[\left(3 - 4 \sin^2 x\right)\left(\sec^2 x - 4 \tan^2 x\right) = \frac{3 - \tan^2 x}{1 - 4 \sin^2 x}\][/tex]
### Step 1: Simplify the Left-hand Side (LHS)
First, let's rewrite the left-hand side:
[tex]\[ \left(3 - 4 \sin^2 x\right) \left(\sec^2 x - 4 \tan^2 x\right) \][/tex]
We know that:
[tex]\[\sec^2 x = 1 + \tan^2 x\][/tex]
So replace [tex]\(\sec^2 x\)[/tex]:
[tex]\[ \left(3 - 4 \sin^2 x\right) \left((1 + \tan^2 x) - 4 \tan^2 x\right) \][/tex]
[tex]\[ \left(3 - 4 \sin^2 x\right) \left(1 + \tan^2 x - 4 \tan^2 x\right) \][/tex]
[tex]\[ \left(3 - 4 \sin^2 x\right) \left(1 - 3 \tan^2 x\right) \][/tex]
### Simplified LHS:
[tex]\[
(3 - 4 \sin^2 x) (1 - 3 \tan^2 x )
\][/tex]
### Step 2: Simplify the Right-hand Side (RHS)
Now consider the right-hand side:
[tex]\[ \frac{3 - \tan^2 x}{1 - 4 \sin^2 x} \][/tex]
### Step 3: Compare the Simplified LHS and RHS
We now have:
- LHS: [tex]\((4 \sin^2 x - 3) \cdot (3 \tan^2 x - 1)\)[/tex]
- RHS: [tex]\(\frac{3 - \tan^2 x}{1 - 4 \sin^2 x} \)[/tex]
From the comparison:
- LHS involves polynomial products, [tex]\((4 \sin^2 x - 3) \cdot (3 \tan^2 x - 1)\)[/tex]
- RHS involves fraction of polynomials, [tex]\(\frac{(3-\tan^2 x)}{(4 \sin^2 x - 1)}\)[/tex]
Since polynomial and fraction involve complex trigonometric functions and their identities, upon reducing, we notice that:
The expressions [tex]\( \left(4 \sin^2 x - 3\right) (3 \tan^2 x - 1) \)[/tex] do not simplify to [tex]\( \frac{3 - \tan^2 x}{1 - 4 \sin^2 x} \)[/tex]
### Conclusion:
Thus, the initial given equation turns out to be false as:
\[
\left(3 - 4 \sin^2 x\right)\left(\sec^2 x - 4 \tan^2 x\right) \neq \frac{\left(3 - \tan^2 x\right)}{\left(1 - 4 \sin^2 x\right)}
