Certainly! Let's solve the given problem step by step:
We need to verify the equation:
[tex]\[ 2 \times \sin 30^{\circ} = \frac{2 \tan 30^{\circ}}{1 + \tan^2 30^{\circ}} \][/tex]
### Step 1: Evaluate [tex]\( \sin 30^{\circ} \)[/tex]
The sine of 30 degrees is a well-known trigonometric value.
[tex]\[
\sin 30^{\circ} = 0.5
\][/tex]
### Step 2: Evaluate [tex]\( \tan 30^{\circ} \)[/tex]
The tangent of 30 degrees is also a well-known trigonometric value.
[tex]\[
\tan 30^{\circ} = \frac{1}{\sqrt{3}} \approx 0.5773502691896257
\][/tex]
### Step 3: Compute the Left Side of the Equation
The left side of the equation is [tex]\( 2 \times \sin 30^{\circ} \)[/tex].
[tex]\[
2 \times \sin 30^{\circ} = 2 \times 0.5 = 1.0
\][/tex]
### Step 4: Compute the Right Side of the Equation
The right side of the equation is [tex]\( \frac{2 \tan 30^{\circ}}{1 + \tan^2 30^{\circ}} \)[/tex].
First, we compute [tex]\( \tan^2 30^{\circ} \)[/tex]:
[tex]\[
\tan^2 30^{\circ} = \left( \frac{1}{\sqrt{3}} \right)^2 = \frac{1}{3} \approx 0.3333333333333333
\][/tex]
Now, add 1 to [tex]\( \tan^2 30^{\circ} \)[/tex]:
[tex]\[
1 + \tan^2 30^{\circ} = 1 + 0.3333333333333333 = 1.3333333333333333
\][/tex]
Then, compute [tex]\( \frac{2 \tan 30^{\circ}}{1 + \tan^2 30^{\circ}} \)[/tex]:
[tex]\[
\frac{2 \times 0.5773502691896257}{1.3333333333333333} = \frac{1.1547005383792514}{1.3333333333333333} \approx 0.8660254037844386
\][/tex]
### Step 5: Compare Both Sides of the Equation
The left side is:
[tex]\[
2 \times \sin 30^{\circ} = 1.0
\][/tex]
The right side is:
[tex]\[
\frac{2 \tan 30^{\circ}}{1 + \tan^2 30^{\circ}} \approx 0.8660254037844386
\][/tex]
### Conclusion
Upon comparing both sides of the equation:
[tex]\[
2 \times \sin 30^{\circ} = 1.0 \quad \text{and} \quad \frac{2 \tan 30^{\circ}}{1 + \tan^2 30^{\circ}} \approx 0.8660254037844386
\][/tex]
We observe that:
[tex]\[
1.0 \neq 0.8660254037844386
\][/tex]
Therefore, the given equation [tex]\( 2 \times \sin 30^{\circ} = \frac{2 \tan 30^{\circ}}{1 + \tan^2 30^{\circ}} \)[/tex] is not true.
