Prove

Cos (x+60)= 2Sinx

I doubt it.

say x = 30 degrees
cos (30+60) = cos 90 = 0
but
2 sin 30 = 1

So I claim that your statement is not true.

To prove the trigonometric identity Cos (x+60) = 2Sin x, we need to simplify both sides of the equation and verify that they are equal.

Let's start by using the angle sum formula for cosine:

Cos (a + b) = Cos a * Cos b - Sin a * Sin b

Using this formula, we can rewrite the left side of the equation:

Cos (x+60) = Cos x * Cos 60 - Sin x * Sin 60

Since Cos 60 = 1/2 and Sin 60 = √3/2, we substitute these values:

Cos (x+60) = Cos x * (1/2) - Sin x * (√3/2)

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

2Sin x = 2 * Sin x

Now we have:

Cos (x+60) = Cos x * (1/2) - Sin x * (√3/2) = 2 * Sin x

To verify that these two sides are equal, we need to simplify further:

Multiply both sides of the equation by 2 to remove the denominator:

2 * Cos (x+60) = Cos x - Sin x * √3

Now, let's use the double-angle formula for cosine to rewrite the left side:

2 * Cos (x+60) = 2 * Cos x * Cos 60 - 2 * Sin x * Sin 60

Since Cos 60 = 1/2 and Sin 60 = √3/2, we substitute these values:

2 * Cos (x+60) = 2 * Cos x * (1/2) - 2 * Sin x * (√3/2)

Simplifying further:

2 * Cos (x+60) = Cos x - Sin x * √3

Comparing this result to the right side of the equation, we see that they are equal:

2 * Cos (x+60) = Cos x - Sin x * √3 = 2 * Sin x

Therefore, we have proven that Cos (x+60) = 2Sin x.