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Question
Solve the following equation:
\[5 \cos^2 x + 7 \sin^2 x - 6 = 0\]
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Solution
\[5 \cos^2 x + 7 \sin^2 x - 6 = 0\]
\[ \Rightarrow 5 \cos^2 x + 7\left( 1 - \cos^2 x \right) - 6 = 0\]
\[ \Rightarrow - 2 \cos^2 x + 1 = 0\]
\[ \Rightarrow \cos^2 x = \frac{1}{2} = \cos^2 \frac{\pi}{4}\]
\[ \Rightarrow x = n\pi \pm \frac{\pi}{4}, n \in Z \left( \cos^2 x = \cos^2 \alpha \Rightarrow x = n\pi \pm \alpha, n \in Z \right)\]
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