Question

Sin^2 x+cos^2 x=sin2x

Answer 1

Ответ:

x=пи/4+пи*н

н - целые числа

Пошаговое объяснение:

Sin^2 x+cos^2 x=sin2x

1=sin2x

sin2x=1

2x=пи/2+2пи*н

x=пи/4+пи*н

н - целые числа

Answer 2

The equation you provided, sin^2(x) + cos^2(x) = sin(2x), is a trigonometric identity known as the Pythagorean identity. It states that for any angle x, the sum of the squares of the sine and cosine functions is equal to the sine of twice that angle.

To prove this identity, we can start with the double-angle identity for sine:

sin(2x) = 2sin(x)cos(x).

Now let's simplify the left-hand side of the equation:

sin^2(x) + cos^2(x)

Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can replace sin^2(x) + cos^2(x) with 1:

1 = 2sin(x)cos(x).

Dividing both sides of the equation by 2sin(x)cos(x), we get:

1/(2sin(x)cos(x)) = 1.

Now, if we multiply the numerator and denominator of the left-hand side by 2, we get:

2/(2sin(x)cos(x)) = 1.

Simplifying further:

1/sin(x)cos(x) = 1.

Recall that the reciprocal of a product is the same as the product of the reciprocals. Therefore:

1/sin(x)cos(x) = csc(x)sec(x),

where csc(x) represents the cosecant of x and sec(x) represents the secant of x.

Finally, we have:

csc(x)sec(x) = 1.

Therefore, sin^2(x) + cos^2(x) is indeed equal to sin(2x), as given by the Pythagorean identity.