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Question
If \[\cos 4x = 1 + k \sin^2 x \cos^2 x\] , then write the value of k.
Solution
\[\text{ We have,} \]
\[ \cos4x = 1 + k \sin^2 x \cos^2 x \]
\[ \Rightarrow \cos\left( 2 \times 2x \right) = 1 + k \sin^2 x \cos^2 x \]
\[ \Rightarrow 1 - 2 \sin^2 2x = 1 + k \sin^2 x \cos^2 x \]
\[ \Rightarrow 1 - 2 \left( 2sinxcosx \right)^2 = 1 + k \sin^2 x \cos^2 x \]
\[ \Rightarrow 1 - 8 \sin^2 x \cos^2 x = 1 + k \sin^2 x \cos^2 x \]
\[ \Rightarrow \sin^2 {xcos}^2 x\left( k + 8 \right) = 0\]
\[ \Rightarrow k + 8 = 0\]
\[ \therefore k = - 8\]
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