Упростите выражение: 1) $1 - 2sin^25\alpha$; 2) $2cos^23\alpha - 1$;

Mathematics
1

Упростите выражение: 1) 1 - 2sin^25\alpha; 2) 2cos^23\alpha - 1; 3) \frac{1 - cos2\alpha}{sin\frac{\alpha}{2}cos\frac{\alpha}{2}}; 4) \frac{2cos^2\frac{\alpha}{2} - 1}{sin2\alpha}; 5) 1 + cos4\alpha; 6) 1 - 2cos^25\alpha.

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To simplify these trigonometric expressions, we’ll use identities such as the double angle formulas and the Pythagorean identity. Let’s divide the work into six parts as indicated by each given expression.

1) Simplifying ( 1 - 2\sin^2 5\alpha )

Step-by-Step Explanation:

We start by using the Pythagorean identity and a double angle identity for cosine:
\cos 2\theta = 1 - 2\sin^2 \theta
This identity allows us to transform the expression to the form of a cosine double angle:
1 - 2\sin^2 5\alpha = \cos 10\alpha

Conclusion:

Thus, the expression simplifies to:
\boxed{\cos 10\alpha}

2) Simplifying 2\cos^2 3\alpha - 1

Step-by-Step Explanation:

We use the trigonometric identity:
\cos 2\theta = 2\cos^2 \theta - 1
Thus:
2\cos^2 3\alpha - 1 = \cos 6\alpha

Conclusion:

The expression simplifies to:
\boxed{\cos 6\alpha}

3) Simplifying \frac{1 - \cos 2\alpha}{\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}}

Step-by-Step Explanation:

The numerator can be simplified using the double angle identity for sine:
1 - \cos 2\alpha = 2\sin^2 \alpha
And remember:
\sin \alpha = 2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}
Substitute back:
\frac{1 - \cos 2\alpha}{\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}} = \frac{2\sin^2 \alpha}{\frac{1}{2}\sin\alpha} = 2\sin\alpha

Conclusion:

The expression simplifies to:
\boxed{2\sin\alpha}

4) Simplifying \frac{2\cos^2\frac{\alpha}{2} - 1}{\sin 2\alpha}

Step-by-Step Explanation:

Using the identity \cos 2\theta = 2\cos^2 \theta - 1 :

2\cos^2\frac{\alpha}{2} - 1 = \cos \alpha

And \sin 2\theta = 2\sin \theta \cos \theta :

\sin 2\alpha = 2\sin\alpha\cos\alpha

Therefore, simplify to:
\frac{\cos \alpha}{2\sin\alpha\cos\alpha} = \frac{1}{2\sin\alpha}

Conclusion:

The expression simplifies to:
\boxed{\frac{1}{2\sin\alpha}}

5) Simplifying ( 1 + \cos 4\alpha )

Step-by-Step Explanation:

Using the identity 1 + \cos 2\theta = 2\cos^2 \theta :

1 + \cos 4\alpha = 2\cos^2 2\alpha

Conclusion:

Thus, the expression simplifies to:
\boxed{2\cos^2 2\alpha}

6) Simplifying 1 - 2\cos^2 5\alpha

Step-by-Step Explanation:

Using the double angle identity:
\cos 2\theta = 2\cos^2 \theta - 1 \Rightarrow 1 - 2\cos^2 \theta = -\cos 2\theta

Thus:
1 - 2\cos^2 5\alpha = -\cos 10\alpha

Conclusion:

The expression simplifies to:
\boxed{-\cos 10\alpha}

Here’s the simplified answer for each:

  1. ** \cos 10\alpha **
  2. ** \cos 6\alpha **
  3. ** 2\sin\alpha **
  4. ** \frac{1}{2\sin\alpha} **
  5. ** 2\cos^2 2\alpha **
  6. ** -\cos 10\alpha **

Feel free to ask if you have any further queries! @username

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