Sure, let's solve the given expression step-by-step.
The expression given is:
[tex]\[ \frac{2 \cos \theta + \sin 4s}{\tan 45^\circ - (\cos 45^\circ + \sin 90^\circ)} \][/tex]
1. Evaluate the Trigonometric Constants:
- [tex]\(\tan 45^\circ\)[/tex]:
[tex]\[\tan 45^\circ = 1 \][/tex]
- [tex]\(\cos 45^\circ\)[/tex]:
[tex]\[\cos 45^\circ = \frac{\sqrt{2}}{2} \][/tex]
- [tex]\(\sin 90^\circ\)[/tex]:
[tex]\[\sin 90^\circ = 1 \][/tex]
2. Substitute these values into the expression:
The denominator becomes:
[tex]\[ \tan 45^\circ - (\cos 45^\circ + \sin 90^\circ) \][/tex]
Substituting the values, we get:
[tex]\[ 1 - \left(\frac{\sqrt{2}}{2} + 1\right) \][/tex]
3. Simplify the Denominator:
[tex]\[ 1 - \left(\frac{\sqrt{2}}{2} + 1\right) = 1 - \frac{\sqrt{2}}{2} - 1 \][/tex]
[tex]\[ = - \frac{\sqrt{2}}{2} \][/tex]
4. Rewrite the Entire Expression:
[tex]\[ \frac{2 \cos \theta + \sin 4s}{- \frac{\sqrt{2}}{2}} \][/tex]
5. Simplify the Fraction:
Dividing by [tex]\(- \frac{\sqrt{2}}{2}\)[/tex] is the same as multiplying by [tex]\(- \frac{2}{\sqrt{2}}\)[/tex]:
[tex]\[ \left(2 \cos \theta + \sin 4s\right) \cdot \left(- \frac{2}{\sqrt{2}}\right) \][/tex]
6. Simplify Further:
[tex]\[ \left(2 \cos \theta + \sin 4s\right) \cdot \left(- \sqrt{2}\right) \][/tex]
Distributing the multiplication, we get:
[tex]\[ - \sqrt{2} \cdot \left(2 \cos \theta + \sin 4s\right) \][/tex]
[tex]\[ = - 2\sqrt{2} \cos \theta - \sqrt{2} \sin 4s \][/tex]
So, the simplified form of the given trigonometric expression is:
[tex]\[ - 2\sqrt{2} \cos \theta - \sqrt{2} \sin 4s \][/tex]
