To find the value of the expression [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], we will break this down into steps by evaluating each trigonometric function separately and then combining the results.
1. Evaluate [tex]\(\tan^2 30^{\circ}\)[/tex]:
- We know that [tex]\(\tan 30^{\circ} = \frac{1}{\sqrt{3}}\)[/tex].
- Therefore, [tex]\(\tan^2 30^{\circ} = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3}\)[/tex].
2. Evaluate [tex]\(\sin^2 60^{\circ}\)[/tex]:
- We know that [tex]\(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\)[/tex].
- Therefore, [tex]\(\sin^2 60^{\circ} = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\)[/tex].
3. Evaluate [tex]\(\tan^2 60^{\circ}\)[/tex]:
- We know that [tex]\(\tan 60^{\circ} = \sqrt{3}\)[/tex].
- Therefore, [tex]\(\tan^2 60^{\circ} = (\sqrt{3})^2 = 3\)[/tex].
4. Evaluate [tex]\(\tan^2 45^{\circ}\)[/tex]:
- We know that [tex]\(\tan 45^{\circ} = 1\)[/tex].
- Therefore, [tex]\(\tan^2 45^{\circ} = 1^2 = 1\)[/tex].
Now, substituting these values back into the given expression:
[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]
This becomes:
[tex]\[
\frac{4}{3} \left( \frac{1}{3} \right) + \left( \frac{3}{4} \right) + \frac{3}{4} (3) - 2 (1)
\][/tex]
Let's evaluate each term step by step:
1. [tex]\(\frac{4}{3} \left( \frac{1}{3} \right)\)[/tex] = [tex]\(\frac{4}{9} = 0.4444444444444444\)[/tex]
2. [tex]\(\sin^2 60^{\circ} = \frac{3}{4} = 0.75\)[/tex]
3. [tex]\(\frac{3}{4} \tan^2 60^{\circ} = \frac{3}{4} (3) = \frac{9}{4} = 2.25\)[/tex]
4. [tex]\(2 \tan^2 45^{\circ} = 2 (1) = 2\)[/tex]
Adding these results together:
[tex]\[
0.4444444444444444 + 0.75 + 2.25 - 2
\][/tex]
Now, compute the sum:
[tex]\[
0.4444444444444444 + 0.75 + 2.25 - 2 = 1.4444444444444444
\][/tex]
Therefore, the value of the expression is [tex]\(\boxed{1.4444444444444444}\)[/tex].
