Let's solve the expression step by step to find the value of [tex]\(\frac{4}{3} \tan^2 30^{\circ} + \sin^2 60^{\circ} + \frac{3}{4} \tan^2 60^{\circ} - 2 \tan^2 45^{\circ}\)[/tex].
1. Calculate [tex]\(\tan^2 30^{\circ}\)[/tex]:
First, remember that [tex]\(\tan 30^{\circ} = \frac{1}{\sqrt{3}}\)[/tex].
[tex]\[
\tan^2 30^{\circ} = \left( \frac{1}{\sqrt{3}} \right)^2 = \frac{1}{3}
\][/tex]
Now calculate [tex]\(\frac{4}{3} \tan^2 30^{\circ}\)[/tex]:
[tex]\[
\frac{4}{3} \tan^2 30^{\circ} = \frac{4}{3} \cdot \frac{1}{3} = \frac{4}{9} \approx 0.444444
\][/tex]
2. Calculate [tex]\(\sin^2 60^{\circ}\)[/tex]:
Recall that [tex]\(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\)[/tex].
[tex]\[
\sin^2 60^{\circ} = \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3}{4} = 0.75
\][/tex]
3. Calculate [tex]\(\tan^2 60^{\circ}\)[/tex]:
We know [tex]\(\tan 60^{\circ} = \sqrt{3}\)[/tex].
[tex]\[
\tan^2 60^{\circ} = (\sqrt{3})^2 = 3
\][/tex]
Now calculate [tex]\(\frac{3}{4} \tan^2 60^{\circ}\)[/tex]:
[tex]\[
\frac{3}{4} \tan^2 60^{\circ} = \frac{3}{4} \cdot 3 = \frac{9}{4} = 2.25
\][/tex]
4. Calculate [tex]\(\tan^2 45^{\circ}\)[/tex]:
Remember that [tex]\(\tan 45^{\circ} = 1\)[/tex].
[tex]\[
\tan^2 45^{\circ} = 1^2 = 1
\][/tex]
Now calculate [tex]\(2 \tan^2 45^{\circ}\)[/tex]:
[tex]\[
2 \tan^2 45^{\circ} = 2 \cdot 1 = 2
\][/tex]
5. Combine all terms:
Now, add the terms together and subtract the last term:
[tex]\[
\frac{4}{3} \tan^2 30^{\circ} + \sin^2 60^{\circ} + \frac{3}{4} \tan^2 60^{\circ} - 2 \tan^2 45^{\circ}
\][/tex]
Substituting the calculated values:
[tex]\[
0.444444 + 0.75 + 2.25 - 2
\][/tex]
Simplify this expression:
[tex]\[
0.444444 + 0.75 + 2.25 - 2 = 1.444444
\][/tex]
So, the value of the given expression is [tex]\(1.444444\)[/tex].
