Question

требуется помощь гения геометрии (задание б)​

Answer 1

The equation you've written is a trigonometric identity that holds true for any angle "a." It's a manifestation of the trigonometric identity known as the sum-to-product identity. Here's a breakdown of the identity you've written:

cos(a) + sin(a)sin(180° - a) = 1

The identity can be proven as follows:

1. Start with the sum-to-product identity for sine:

sin(A)sin(B) = 1/2[cos(A - B) - cos(A + B)]

2. Substitute A = a and B = (180° - a):

sin(a)sin(180° - a) = 1/2[cos(a - (180° - a)) - cos(a + (180° - a))]

3. Simplify the expressions inside the brackets:

cos(a - (180° - a)) = cos(a + a) = cos(2a)
cos(a + (180° - a)) = cos(a + a) = cos(2a)

4. Substitute these simplified expressions back into the identity:

1/2[cos(2a) - cos(2a)]

5. Since we are subtracting cos(2a) from itself, it equals zero:

1/2[0] = 0

6. Now, we have:

sin(a)sin(180° - a) = 0

7. Add cos(a) to both sides of the equation:

cos(a) + sin(a)sin(180° - a) = cos(a) + 0 = cos(a) = 1

So, the trigonometric identity cos(a) + sin(a)sin(180° - a) = 1 is true for any angle "a."