Sure! Let's solve the given equation step by step:
### Step 1: Find the individual trigonometric values:
1. [tex]\(\sin 45^\circ\)[/tex]:
[tex]\[
\sin 45^\circ = \frac{\sqrt{2}}{2} \approx 0.707 \implies \sin^2 45^\circ = \left( \frac{\sqrt{2}}{2} \right)^2 = \frac{1}{2} \approx 0.5
\][/tex]
2. [tex]\(\cos 30^\circ\)[/tex]:
[tex]\[
\cos 30^\circ = \frac{\sqrt{3}}{2} \approx 0.866 \implies \cos^2 30^\circ = \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3}{4} \approx 0.75
\][/tex]
3. [tex]\(\csc 45^\circ\)[/tex]:
[tex]\[
\csc 45^\circ = \frac{1}{\sin 45^\circ} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \approx 1.414 \implies \csc^2 45^\circ = \left( \sqrt{2} \right)^2 = 2 \approx 2.0
\][/tex]
4. [tex]\(\sec 60^\circ\)[/tex]:
[tex]\[
\sec 60^\circ = \frac{1}{\cos 60^\circ} = \frac{1}{\frac{1}{2}} = 2 \approx 2.0 \implies \sec^2 60^\circ = \left( 2 \right)^2 = 4 \approx 4.0
\][/tex]
### Step 2: Substitute these values into the given equation:
The equation is:
[tex]\[
\tan A + \sin^2 45^\circ - \cos^2 30^\circ - \csc^2 45^\circ \sec^2 60^\circ = 4
\][/tex]
Substituting in the numerical values:
[tex]\[
\tan A + 0.5 - 0.75 - (2 \times 4) = 4
\][/tex]
### Step 3: Simplify the expression step by step:
1. Calculate [tex]\(\csc^2 45^\circ \sec^2 60^\circ\)[/tex]:
[tex]\[
2 \times 4 = 8
\][/tex]
2. Combine the terms in the equation:
[tex]\[
\tan A + 0.5 - 0.75 - 8 = 4
\][/tex]
3. Further simplify:
[tex]\[
\tan A + 0.5 - 0.75 - 8 = 4
\][/tex]
[tex]\[
\tan A - 8.25 = 4
\][/tex]
### Step 4: Solve for [tex]\(\tan A\)[/tex]:
[tex]\[
\tan A = 4 + 8.25
\][/tex]
[tex]\[
\tan A = 12.25
\][/tex]
So, the value of [tex]\(\tan A\)[/tex] is [tex]\(12.25\)[/tex].
