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Question
Write the value of 2 (sin6 x + cos6 x) −3 (sin4 x + cos4 x) + 1.
Solution
\[2\left( \sin^6 x + \cos^6 x \right) - 3\left( \sin^4 x + \cos^4 x \right) + 1\]
\[ = 2\left( \sin^2 x + \cos^2 x \right)\left( \sin^4 x + \cos^4 x - \sin^2 x . c {os}^2 x \right) - 3\left( \sin^4 x + \cos^4 x \right) + 1\]
\[ = 2 . 1\left( \sin^4 x + \cos^4 x - \sin^2 x . c {os}^2 x \right) - 3\left( \sin^4 x + \cos^4 x \right) + 1\]
\[ = 2\left( \sin^4 x + \cos^4 x \right) - 2 \sin^2 x . c {os}^2 x - 3\left( \sin^4 x + \cos^4 x \right) + 1\]
\[ = - \left( \sin^4 x + \cos^4 x \right) - 2 \sin^2 x . c {os}^2 x + 1\]
\[ = - \left\{ \sin^4 x + \cos^4 x + 2 \sin^2 x . c {os}^2 x \right\} + 1\]
\[ = - \left( \sin^2 x + \cos^2 x \right)^2 + 1\]
\[ = - 1 + 1\]
\[ = 0\]
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