मराठी

Prove That: 1 + Cos 2 2 X = 2 ( Cos 4 X + Sin 4 X )

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प्रश्न

Prove that: \[1 + \cos^2 2x = 2 \left( \cos^4 x + \sin^4 x \right)\]

 
संख्यात्मक
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उत्तर

\[LHS = 1 + \cos^2 2x\]

Using the identity

\[\cos2x = \cos^2 x - s {in}^2 x\], we get

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

\[ = 1 + \cos^4 x + \sin^4 x - 2 \cos^2 x \sin^2 x\]

\[= \left( \cos^2 x + \sin^2 x \right)^2 + \cos^4 x + \sin^4 x - 2 \cos^2 x \sin^2 x \left[ \because \cos^2 x + \sin^2 x = 1 \right]\]

\[ = \cos^4 x + \sin^4 x + 2 \cos^2 x \sin^2 x + \cos^4 x + \sin^4 x - 2 \cos^2 x \sin^2 x\]

\[ = 2( \cos^4 x + \sin^4 x) = RHS\]

\[\text{ Hence proved } .\]

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Values of Trigonometric Functions at Multiples and Submultiples of an Angle
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पाठ 9: Values of Trigonometric function at multiples and submultiples of an angle - Exercise 9.1 [पृष्ठ २८]

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आर.डी. शर्मा Mathematics [English] Class 11
पाठ 9 Values of Trigonometric function at multiples and submultiples of an angle
Exercise 9.1 | Q 13 | पृष्ठ २८

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