Sin 2x Cos 2x – Value, Calculation, Formula, Methods

Value of Sin 2x Cos 2x

The value of sin 2x cos 2x is 1.

Derivation of Sin 2x Cos 2x

We start with the identity sin2x + cos2x = 1.

Then we use the Pythagorean theorem to write:

sin2x = (sin x)2

cos2x = (cos x)2

Now we can subtract the two equations:

sin2x – cos2x = (sin x)2 – (cos x)2

= (sin x)2 – (cos x)2

= (1 – cos 2x)

Step wise calculation

The value of sin 2x can be calculated using the formula:

sin 2x = 2 sin x cos x

Similarly, the value of cos 2x can be calculated using the formula:

cos 2x = cos²x – sin²x

We can substitute the value of sin 2x and cos 2x in terms of sin x and cos x in the expression sin 2x cos 2x as follows:

sin 2x cos 2x = (2 sin x cos x) (cos²x – sin²x)

= 2 sin x cos x cos²x – 2 sin x cos x sin²x

= 2 sin x cos³x – 2 sin³x cos x

= 2 sin x cos x (cos²x – sin²x)

= 2 sin x cos x cos 2x

Therefore, we can calculate the value of sin 2x cos 2x by first calculating the value of cos 2x using the formula cos 2x = cos²x – sin²x, and then substituting the value of cos 2x in the expression 2 sin x cos x cos 2x.

Here are the step-wise calculations:

1. Calculate cos 2x using the formula cos 2x = cos²x – sin²x.

2. Substitute the value of cos 2x in the expression 2 sin x cos x cos 2x.

3. Simplify the expression using trigonometric identities.

Note that the final answer will depend on the value of x, as well as the units used to measure angles (radians or degrees).