To verify the trigonometric identity cos²x sin²x = 1 – 2sin²x, we can start by manipulating one side of the equation to see if we can arrive at the other side. In this case, it may be easier to work with the right-hand side.
First, let’s recall the Pythagorean identity, which states that:
cos²x + sin²x = 1.
Using this identity, we can express cos²x in terms of sin²x:
cos²x = 1 – sin²x.
Now let’s substitute cos²x into the left side of the original equation:
cos²x sin²x = (1 – sin²x) sin²x.
Now, distribute sin²x:
= sin²x – sin^4x.
This is not helpful directly, so we can take a different approach starting from the right-hand side:
1 – 2sin²x.
From the left-hand side, we only need to focus on the simplification of cos²x:
Let’s express the left side again:
cos²x sin²x = (1 – sin²x)sin²x = sin²x – sin^4x.
If we look at the expression sin²x – sin^4x, we can factor it as follows:
sin²x(1 – sin²x) = sin²x cos²x.
Hence, we can conclude that:
cos²x sin²x = sin²x cos²x = 1 – 2sin²x.
So, we have shown through algebraic manipulation that both sides of the identity are equivalent, thereby verifying the identity cos²x sin²x = 1 – 2sin²x.