Question

show that:
sin4theta + cos4theta = 1- cos2theta.

Answer 1

Solution :

Here I am using A instead of theta.

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We know the algebraic identity:

a² + 2ab + b² = ( a + b )²

Or

a² + b² = ( a + b )² - 2ab

and

By Trigonometric identity:

sin²A + cos²A = 1

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Here ,

LHS = sin⁴ A + cos⁴A

= ( sin²A )² + ( cos²A )²

= ( sin²A + cos²A )² -2sin²Acos²A

= 1 - 1/2( 4sin²Acos²A )

= 1 - 1/2 ( sin²2A )

= ( 2 - sin²2A )/2

= ( 1 + 1 - sin²2A )/2

= ( 1 + cos²2A )/2

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