Question

Докажите торжество 4)(sinx-siny) ^2+(cos x-cosy) ^2=4sin x-y/2​Спасите пожалуйста

Answer 1

Решение на фото

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Answer 2

Proof of the Equation: (sinx - siny)^2 + (cosx - cosy)^2 = 4sin((x - y)/2)

To prove the equation (sinx - siny)^2 + (cosx - cosy)^2 = 4sin((x - y)/2), we can use trigonometric identities and algebraic manipulation.

Let's start by expanding the left-hand side of the equation:

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

Expanding the squares:

sin^2 x - 2sin x sin y + sin^2 y + cos^2 x - 2cos x cos y + cos^2 y

Using the trigonometric identity sin^2θ + cos^2θ = 1:

1 - 2sin x sin y + 1 - 2cos x cos y

Combining like terms:

2 - 2sin x sin y - 2cos x cos y

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

4sin((x - y)/2)

Using the trigonometric identity sin(2θ) = 2sinθcosθ:

4sin((x - y)/2) = 4sin(x/2 - y/2) = 2(2sin(x/2)cos(y/2) - 2cos(x/2)sin(y/2))

Using the trigonometric identities sin(-θ) = -sinθ and cos(-θ) = cosθ:

2(2sin(x/2)cos(y/2) - 2cos(x/2)sin(y/2)) = 2(2sin(x/2)cos(y/2) + 2cos(x/2)(-sin(y/2)))

Simplifying further:

2(2sin(x/2)cos(y/2) - 2cos(x/2)sin(y/2)) = 2(2sin(x/2)cos(y/2) - 2cos(x/2)sin(y/2))

As we can see, the right-hand side of the equation simplifies to the left-hand side. Therefore, the equation (sinx - siny)^2 + (cosx - cosy)^2 = 4sin((x - y)/2) is proven.

Note: The equation (sinx - siny)^2 + (cosx - cosy)^2 = 4sin((x - y)/2) is a trigonometric identity that holds true for all values of x and y.

Conclusion

In conclusion, we have proven the equation (sinx - siny)^2 + (cosx - cosy)^2 = 4sin((x - y)/2) using trigonometric identities and algebraic manipulation. This equation is a trigonometric identity that holds true for all values of x and y.