Answer:
Easy
Explanation:
To prove the given equation:
\[ 4(\sin^4 60^\circ + \cos^2 60^\circ) - 3(\tan^2 60^\circ - \tan^2 45^\circ) + 5(\cos^2 45^\circ) = -\frac{1}{4} \]
Let's break it down step by step:
1. **Calculate each term individually:**
- **For \( \sin^4 60^\circ \) and \( \cos^2 60^\circ \):**
First, determine \( \sin 60^\circ \) and \( \cos 60^\circ \):
\[ \sin 60^\circ = \frac{\sqrt{3}}{2}, \quad \cos 60^\circ = \frac{1}{2} \]
Therefore,
\[ \sin^2 60^\circ = \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3}{4}, \quad \cos^2 60^\circ = \left( \frac{1}{2} \right)^2 = \frac{1}{4} \]
Now, compute \( \sin^4 60^\circ \):
\[ \sin^4 60^\circ = \left( \sin^2 60^\circ \right)^2 = \left( \frac{3}{4} \right)^2 = \frac{9}{16} \]
And \( \cos^2 60^\circ \):
\[ \cos^2 60^\circ = \frac{1}{4} \]
- **For \( \tan 60^\circ \) and \( \tan 45^\circ \):**
\[ \tan 60^\circ = \sqrt{3}, \quad \tan 45^\circ = 1 \]
Compute \( \tan^2 60^\circ \):
\[ \tan^2 60^\circ = (\sqrt{3})^2 = 3 \]
And \( \tan^2 45^\circ \):
\[ \tan^2 45^\circ = 1 \]
- **For \( \cos 45^\circ \):**
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]
Compute \( \cos^2 45^\circ \):
\[ \cos^2 45^\circ = \left( \frac{\sqrt{2}}{2} \right)^2 = \frac{2}{4} = \frac{1}{2} \]
2. **Substitute these values into the original equation:**
Substitute \( \sin^4 60^\circ = \frac{9}{16} \), \( \cos^2 60^\circ = \frac{1}{4} \), \( \tan^2 60^\circ = 3 \), \( \tan^2 45^\circ = 1 \), and \( \cos^2 45^\circ = \frac{1}{2} \) into the equation:
\[ 4(\frac{9}{16} + \frac{1}{4}) - 3(3 - 1) + 5(\frac{1}{2}) \]
3. **Simplify each term:**
\[ 4 \left( \frac{9}{16} + \frac{4}{16} \right) - 3 \cdot 2 + \frac{5}{2} \]
\[ 4 \cdot \frac{13}{16} - 6 + \frac{5}{2} \]
4. **Further simplify:**
\[ \frac{52}{16} - 6 + \frac{5}{2} \]
\[ \frac{52}{16} = \frac{13}{4}, \quad -6 = -\frac{24}{4}, \quad \frac{5}{2} = \frac{10}{4} \]
\[ \frac{13}{4} - \frac{24}{4} + \frac{10}{4} \]
\[ = \frac{13 - 24 + 10}{4} \]
\[ = \frac{-1}{4} \]
Therefore, we have shown that:
\[ 4(\sin^4 60^\circ + \cos^2 60^\circ) - 3(\tan^2 60^\circ - \tan^2 45^\circ) + 5(\cos^2 45^\circ) = -\frac{1}{4} \]
Hence, the equation is proved.
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