Alright, let's break down the given expression [tex]\(3 \tan ^2 45^\circ - \sin ^2 60^\circ - \cot ^2 30^\circ + \sec ^2 45^\circ\)[/tex] step by step.
1. Compute [tex]\(\tan^2 45^\circ\)[/tex]:
- The trigonometric value [tex]\(\tan 45^\circ = 1\)[/tex].
- Therefore, [tex]\(\tan^2 45^\circ = 1^2 = 1\)[/tex].
2. Compute [tex]\(\sin^2 60^\circ\)[/tex]:
- The trigonometric value [tex]\(\sin 60^\circ = \frac{\sqrt{3}}{2}\)[/tex].
- Therefore, [tex]\(\sin^2 60^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \approx 0.75\)[/tex].
3. Compute [tex]\(\cot^2 30^\circ\)[/tex]:
- The trigonometric value [tex]\(\cot 30^\circ = \sqrt{3}\)[/tex].
- Therefore, [tex]\(\cot^2 30^\circ = (\sqrt{3})^2 = 3\)[/tex].
4. Compute [tex]\(\sec^2 45^\circ\)[/tex]:
- The trigonometric value [tex]\(\sec 45^\circ = \sqrt{2}\)[/tex].
- Therefore, [tex]\(\sec^2 45^\circ = (\sqrt{2})^2 = 2\)[/tex].
Now that we have these values, plug them into the original expression:
[tex]\[
3 \tan^2 45^\circ - \sin^2 60^\circ - \cot^2 30^\circ + \sec^2 45^\circ
\][/tex]
Substituting in the values we've calculated:
[tex]\[
= 3 \times 1 - 0.75 - 3 + 2
\][/tex]
Perform the arithmetic step by step:
[tex]\[
= 3 - 0.75 - 3 + 2
\][/tex]
[tex]\[
= 3 - 3 - 0.75 + 2
\][/tex]
[tex]\[
= 0 - 0.75 + 2
\][/tex]
[tex]\[
= -0.75 + 2
\][/tex]
[tex]\[
= 1.25
\][/tex]
Hence, the result of the expression [tex]\(3 \tan^2 45^\circ - \sin^2 60^\circ - \cot^2 30^\circ + \sec^2 45^\circ\)[/tex] is [tex]\(\boxed{1.25}\)[/tex].
