Question

(sin^4 a - cos^4 a + cos^2a) / (2*(1-cosa))

Answer 1

$\frac{\sin^4 \alpha -\cos^4 \alpha +\cos2 \alpha }{2(1-\cos \alpha )}=\frac{\sin^4 \alpha -\cos^4 \alpha +\cos^2 \alpha-\sin^2 \alpha }{2(1-\cos \alpha )}=\\\\=\frac{\sin^2 \alpha (\sin^2 \alpha -1)+\cos^2 \alpha (1-\cos^2 \alpha )}{2(1-\cos \alpha )}=\frac{\sin^2 \alpha (\sin^2 \alpha -1)+\cos^2 \alpha \sin^2 \alpha }{2(1-\cos \alpha )}=\\\\=\frac{\sin^2 \alpha (\sin^2 \alpha -1+\cos^2 \alpha)}{2(1-\cos \alpha )}=\frac{\sin^2 \alpha (1 -1)}{2(1-\cos \alpha )}=0.$

Answer 2

To simplify the expression (sin^4 a - cos^4 a + cos^2a) / (2*(1-cosa)), let's break it down step by step:

Step 1: Rewrite the numerator using the identity sin^2(a) = 1 - cos^2(a):

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sin^4(a) - cos^4(a) + cos^2(a)
= (sin^2(a))^2 - cos^4(a) + cos^2(a)
= (1 - cos^2(a))^2 - cos^4(a) + cos^2(a)

Step 2: Expand the squared term and simplify:

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(1 - 2cos^2(a) + cos^4(a)) - cos^4(a) + cos^2(a)
= 1 - 2cos^2(a) + cos^4(a) - cos^4(a) + cos^2(a)
= 1 - 2cos^2(a) + cos^2(a)
= 1 - cos^2(a)
= sin^2(a)

Step 3: Simplify the denominator:

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2 * (1 - cos(a))
= 2 - 2cos(a)

Combining the simplified numerator and denominator, we have:

arduinoCopy code
sin^2(a) / (2 - 2cos(a))

And that is the simplified form of the expression (sin^4(a) - cos^4(a) + cos^2(a)) / (2 * (1 - cos(a))).