Short Note

If \[\cos 4x = 1 + k \sin^2 x \cos^2 x\] , then write the value of *k*.

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#### Solution

\[\text{ We have,} \]

\[ \cos4x = 1 + k \sin^2 x \cos^2 x \]

\[ \Rightarrow \cos\left( 2 \times 2x \right) = 1 + k \sin^2 x \cos^2 x \]

\[ \Rightarrow 1 - 2 \sin^2 2x = 1 + k \sin^2 x \cos^2 x \]

\[ \Rightarrow 1 - 2 \left( 2sinxcosx \right)^2 = 1 + k \sin^2 x \cos^2 x \]

\[ \Rightarrow 1 - 8 \sin^2 x \cos^2 x = 1 + k \sin^2 x \cos^2 x \]

\[ \Rightarrow \sin^2 {xcos}^2 x\left( k + 8 \right) = 0\]

\[ \Rightarrow k + 8 = 0\]

\[ \therefore k = - 8\]

Concept: Values of Trigonometric Functions at Multiples and Submultiples of an Angle

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