Let's solve the given expression step-by-step:
[tex]\[
\frac{\sin 104^{\circ}\left(2 \cos ^2 15^{\circ}-1\right)}{\tan 38^{\circ} \cdot \sin ^2 412^{\circ}}
\][/tex]
### Step 1: Calculate the individual trigonometric functions
1. Sin 104°:
[tex]\[
\sin 104^{\circ} \approx 0.9703
\][/tex]
2. Cos² 15°:
[tex]\[
\cos 15^{\circ} \approx 0.2588 \quad \text{(use cosine value here)}
\][/tex]
[tex]\[
\cos^2 15^{\circ} \approx 0.9330
\][/tex]
[tex]\[
2 \cos^2 15^{\circ} - 1 = 2 \times 0.9330 - 1 \approx 0.8660
\][/tex]
3. Tan 38°:
[tex]\[
\tan 38^{\circ} \approx 0.7813
\][/tex]
4. Sin² 412°:
First, note that [tex]\( 412^{\circ} = 412^{\circ} - 360^{\circ} = 52^{\circ} \)[/tex].
[tex]\[
\sin 52^{\circ} \approx 0.7880
\][/tex]
[tex]\[
\sin^2 52^{\circ} \approx 0.6210
\][/tex]
### Step 2: Calculate the numerator and denominator
1. Numerator:
[tex]\[
\sin 104^{\circ} \times (2 \cos^2 15^{\circ} - 1) = 0.9703 \times 0.8660 \approx 0.8403
\][/tex]
2. Denominator:
[tex]\[
\tan 38^{\circ} \times \sin^2 412^{\circ} = 0.7813 \times 0.6210 \approx 0.4851
\][/tex]
### Step 3: Calculate the final result by dividing the numerator by the denominator
[tex]\[
\frac{\text{Numerator}}{\text{Denominator}} = \frac{0.8403}{0.4851} \approx 1.7321
\][/tex]
Thus, the value of the given expression is approximately:
[tex]\[
\boxed{1.7321}
\][/tex]
