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.