Sin^4α-cos^4α /(sinα * cosα)^2

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вопрос

First, let's simplify the numerator, which is sin^4α - cos^4α.

Using the identity a^2 - b^2 = (a + b)(a - b), we can rewrite sin^4α - cos^4α as (sin^2α + cos^2α)(sin^2α - cos^2α).

Since sin^2α + cos^2α = 1 (from the Pythagorean identity), we have (1)(sin^2α - cos^2α).

Next, note that sin^2α - cos^2α can be rewritten in terms of sinα * cosα using the identity sin^2α - cos^2α = -sin(2α). Therefore, the final simplified numerator is -sin(2α).

Now, let's simplify the denominator, which is (sinα * cosα)^2.

Since (sinα cosα)^2 = sin^2α cos^2α, we can use the Pythagorean identity sin^2α + cos^2α = 1 to rewrite sin^2α cos^2α as 1/4 (since sin^2α cos^2α = 1/4).

Therefore, the simplified expression is -sin(2α) / 1/4, which simplifies to -4sin(2α).