Let's work through the solution step by step.
1. Calculate [tex]\(\tan 45^\circ\)[/tex]:
We know that [tex]\(\tan 45^\circ = 1\)[/tex].
2. Calculate [tex]\(\cos 30^\circ\)[/tex]:
The value of [tex]\(\cos 30^\circ\)[/tex] is given by the trigonometric table as [tex]\(\cos 30^\circ = \frac{\sqrt{3}}{2}\)[/tex].
3. Calculate [tex]\(\sin 60^\circ\)[/tex]:
Similarly, [tex]\(\sin 60^\circ\)[/tex] is given by [tex]\(\sin 60^\circ = \frac{\sqrt{3}}{2}\)[/tex].
4. Multiply [tex]\(2\)[/tex] by [tex]\(\tan 45^\circ\)[/tex]:
So, [tex]\(2 \times \tan 45^\circ = 2 \times 1 = 2\)[/tex].
5. Sum and subtract the trigonometric values:
Now let's put it all together:
[tex]\[
2 \tan 45^\circ + \cos 30^\circ - \sin 60^\circ
\][/tex]
Substituting the values we calculated:
[tex]\[
= 2 \times 1 + \frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2}
\][/tex]
Simplify the expression inside the equation:
[tex]\[
= 2 + \frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2}
\][/tex]
6. Final Simplification:
Since [tex]\(\frac{\sqrt{3}}{2}\)[/tex] and [tex]\(\frac{\sqrt{3}}{2}\)[/tex] cancel each other out:
[tex]\[
= 2
\][/tex]
So, the complete and final solution is:
[tex]\[
2 \tan 45^\circ + \cos 30^\circ - \sin 60^\circ = 2
\][/tex]
