KS asked in Science & MathematicsMathematics · 1 decade ago

TRIGONOMETRY QUESTION!!!!!!!!?

Prove that sin^4 X + cos^4 X = 1/4( 3 + cos 4X).

Hence, or otherwise, find the largest possible value and the smallest possible value of sin^4 X + cos^4 X.

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  • Hemant
    Lv 7
    1 decade ago
    Favorite Answer

    ... LHS

    = sin⁴ x + cos⁴ x

    = ( sin² x )² + ( cos² x )² ... Now use : a²+b² = (a+b)² - 2ab

    = ( sin² x + cos² x )² - 2 sin² x cos² x

    = (1)² - (1/2) ( 4 sin² x cos² x )

    = 1 - (1/2) ( 2 sin x cos x )²

    = 1 - (1/2) sin² 2x

    = 1 - (1/2) · (1/2)( 1 - cos 4x )

    = 1 - [ ( 1 - cos 4x ) / 4 ]

    = [ 4 - ( 1 - cos 4x ) ] / 4

    = (1/4)( 3 + cos 4x )

    = RHS ................................. QED.

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    Let : ƒ(x) = sin⁴ x + cos⁴ x = (1/4)( 3 + cos 4x ) = (3/4) + (1/4) cos 4x.

    For all real values of x,

    ... -1 ≤ cos 4x ≤ 1

    ∴ -1(1/4) ≤ (1/4) cos 4x ≤ 1(1/4)

    ∴ (3/4)-(1/4) ≤ [ (3/4) + (1/4) cos 4x ] ≤ (3/4)+(1/4)

    ∴ (1/2) ≤ ƒ(x) ≤ 1

    ∴ Minimum of ƒ(x) = 1/2, ... Maximum of ƒ(x) = 1 ....................... Ans.

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    Happy To Help !

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