(sin^2t 2sint cost-cos^2t^2)=1-sin4t

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вопрос

To simplify the given expression:

(sin^2t)(2sin(t)cos(t)) - cos^2(t)^2
= 2(sin(t)cos(t))^2 - cos^2(t)^2
= 2(sin^2(t)cos^2(t)) - cos^2(t)^2
= 2(sin^2(t)(1-sin^2(t))) - cos^2(t)^2
= 2sin^2(t) - 2sin^4(t) - cos^2(t)^2
= 2sin^2(t) - 2sin^4(t) - (1-sin^2(t))^2
= 2sin^2(t) - 2sin^4(t) - (1 - 2sin^2(t) + sin^4(t))
= 2sin^2(t) - 2sin^4(t) - 1 + 2sin^2(t) - sin^4(t)
= 4sin^2(t) - 3sin^4(t) - 1
= sin(4t) - 1

Therefore, the given expression simplifies to 1 - sin(4t).