Question 360885
Work with the right side only:
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{{{Cos^4x+Sin^4x = 1-expr(1/2)Sin^2(2x)}}}
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Use the identity: {{{red(Sin(2theta)=2Sin(theta)Cos(theta))}}}
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{{{Cos^4x+Sin^4x = 1-expr(1/2)(2Sin(x)Cos(x))^2}}}
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Square the expression in parentheses:
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{{{Cos^4x+Sin^4x = 1-expr(1/2)(2^2Sin^2x*Cos^2x)}}}
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Cancel the 2 into the 2²:
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{{{Cos^4x+Sin^4x = 1-expr(1/cross(2))(2^cross(2)Sin^2x*Cos^2x)}}}
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{{{Cos^4x+Sin^4x = 1-(2Sin^2x*Cos^2x)}}}
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This is a cool trick here. Replace 1 by 1²
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{{{Cos^4x+Sin^4x = 1^2-(2Sin^2x*Cos^2x)}}}
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Replace the 1 using the identity: {{{red(Sin^2theta + Cos^2theta = 1)}}}
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{{{Cos^4x+Sin^4x = (Sin^2x+Cos^2x)^2-(2Sin^2x*Cos^2x)}}}
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Square out the first term on the right:
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{{{Cos^4x+Sin^4x = (Sin^4x+2Sin^2xCos^2x+Cos^4x)-(2Sin^2x*Cos^2x)}}}
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Remove the parentheses:
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{{{Cos^4x+Sin^4x = Sin^4x+2Sin^2xCos^2x+Cos^4x-2Sin^2x*Cos^2x}}}
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Cancel the terms that cancel:
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{{{Cos^4x+Sin^4x = Sin^4x+cross(2Sin^2xCos^2x)+Cos^4x-cross(2Sin^2x*Cos^2x)}}}
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{{{Cos^4x+Sin^4x = Sin^4x+Cos^4x}}}
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Edwin