Hi. Can someone help me with this?
Simplify:
(sin^2x)(cos^2x)+(sin^4x)
The answer is sin^2x, but I have no idea how to get there!
Thanks!
cos^2 x = 1 - sin^2 x
so
sin^2 x [ 1 - sin^2 x ] + sin^4 x
I think you can see what happens now :)
Or....
take out a common factor
sin^2 x(cos^2 x + sin^2 x)
= (sin^2 x)(1) and there it is.
thanks
Certainly! I can help you simplify the given expression.
To simplify (sin^2x)(cos^2x) + (sin^4x), we can use the trigonometric identity called the Pythagorean identity, which states that sin^2x + cos^2x = 1.
Let's start by substituting sin^2x + cos^2x = 1 into the expression. We get:
(sin^2x)(cos^2x) + (sin^4x) = (1 - cos^2x)(cos^2x) + (sin^4x)
To simplify further, let's expand the first term:
= cos^2x - cos^4x + sin^4x
Now, let's rearrange the terms:
= sin^4x - cos^4x + cos^2x
Notice that sin^4x - cos^4x can be further simplified using another trigonometric identity, which is sin^2x - cos^2x = -cos(2x).
Applying this identity, we get:
= -cos(2x) + cos^2x
Since cos^2x = 1 - sin^2x (using the Pythagorean identity again), we can substitute it into the expression:
= -cos(2x) + 1 - sin^2x
Now, combining like terms, we have:
= 1 - sin^2x - cos(2x)
Finally, after rearranging the terms, we get sin^2x as the simplified expression:
= sin^2x
Therefore, the simplified form of (sin^2x)(cos^2x) + (sin^4x) is sin^2x.
I hope this explanation helps you understand the steps to simplify the given expression.