Question

Sin 2(t-3П/2)cos 2(2П-t) : tg 2(t-п/2)cos 2(t-3п/2)

Answer 1

$\frac{sin^2(t-\frac{3\pi}{2})\, cos^2(2\pi -t)}{tg^2(t-\frac{\pi }{2})\, cos^2(t-\frac{3\pi }{2})}=\frac{cos^2t\cdot cos^2t}{ctg^2t\cdot sin^2t}=\frac{cos^4t}{\frac{cos^2t}{sin^2t}\cdot sin^2t}=cos^2t$

Answer 2

To simplify the given expression, let's break it down step by step:

1. Start with the expression: sin(2(t-3π/2)) * cos(2π - t).
2. Use the trigonometric identities:
   • sin(α - β) = sin α * cos β - cos α * sin β
   • cos(2π - θ) = -cos θ
3. Apply the identities to the expression:

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sin(2(t - 3π/2)) * cos(2π - t)
= sin(2t - 3π) * (-cos(t))

4. Further simplify by using the trigonometric identity:
   • sin(θ) = -sin(θ + π)
5. Apply the identity to the expression:

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= -sin(2t - 3π + π) * cos(t)
= -sin(2t - 2π) * cos(t)
= -sin(2(t - π)) * cos(t)
= -(-sin(2(t - π + π/2)))
= sin(2(t - π - π/2))
= sin(2(t - 3π/2))

Now, let's work on the second part of the expression: tan(2(t-π/2)) * cos(2(t-3π/2)).

1. Use the trigonometric identity:
   • tan(α) = sin(α) / cos(α)
2. Apply the identity to the expression:

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tan(2(t - π/2)) * cos(2(t - 3π/2))
= (sin(2(t - π/2)) / cos(2(t - π/2))) * cos(2(t - 3π/2))

3. Cancel out cos(2(t - π/2)) from the numerator and denominator:

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= sin(2(t - π/2))

4. Use the trigonometric identity:
   • sin(θ - π/2) = cos(θ)
5. Apply the identity to the expression:

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= cos(2t - π)

So, the final simplified expression is:

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sin(2(t - 3π/2)) : tg(2(t - π/2)) * cos(2(t - 3π/2))
= sin(2(t - 3π/2)) : cos(2t - π)

Keep in mind that the expression might still be further simplified based on the context or the specific values of t.