Question

1) sin²45° – sin²30° = cos60° – cos-45°.​

Answer 1

Ответ:

1) sin30° cos60° - sin²45° = 1/2•1/2 - 1/2 = -1/4

2) 2cos180° - sin²270° = 2•(-1) - 1 = -3

3) cos30° sin60° + cos² 45° = 1/2•1/2 + 1/2 = 1/4 + 1/2 = 3/4.

4) 3sin 90° - cos² 0° = 3•1 - 1 = 2

Объяснение:

Answer 2

Let's simplify the expression on both sides step by step:

1. Start with the left side of the equation:

   sin²45° - sin²30°

2. Use the trigonometric identity for the difference of squares:

   sin²A - sin²B = (sinA + sinB)(sinA - sinB)

   Applying this identity to your expression:

   (sin45° + sin30°)(sin45° - sin30°)

3. Now, calculate the values of sin45° and sin30°. You can use the values from common trigonometric knowledge:

   sin45° = √2 / 2 sin30° = 1 / 2

4. Plug in these values:

   (√2 / 2 + 1 / 2)(√2 / 2 - 1 / 2)

5. Now, simplify each factor:

   (√2 / 2 + 1 / 2) = (1 + √2) / 2 (√2 / 2 - 1 / 2) = (√2 - 1) / 2

6. Multiply these two simplified factors together:

   ((1 + √2) / 2) * ((√2 - 1) / 2)

7. Multiply the numerators and the denominators:

   (1 + √2) * (√2 - 1) / (2 * 2)

8. Use the difference of squares identity again:

   (a - b)(a + b) = a² - b²

   Applying this identity to the expression:

   (1 + √2) * (√2 - 1) = (1 + √2)² - (1)²

9. Expand the squares and subtract:

   (1 + 2√2 + 2) - 1 = 2√2 + 2

Now, let's simplify the right side of the equation:

cos60° - cos(-45°)

1. Calculate the values of cos60° and cos(-45°):

   cos60° = 1/2 cos(-45°) = cos(45°) [cosine function is even] = √2 / 2

2. Subtract these values:

   1/2 - √2 / 2

3. Find a common denominator:

   (1 - √2) / 2

Now, comparing the left and right sides of the equation:

Left side: 2√2 + 2 Right side: (1 - √2) / 2

These two sides are not equal, so the given equation is not true:

2√2 + 2 ≠ (1 - √2) / 2