Question

verify the identities.
sin^4 x- cos^4 x= 2sin^2 x- 1

Answer 1

Task:

Verify the identities.

$$\sin^ {4} x - \cos^ {4} x = 2 \sin^ {2} x - 1.$$

Solution:

Let's transform the left part of the expression:

We use the relationship $x^{2} - y^{2} = (x + y)(x - y)$ [difference between two squares]:

$$\sin^ {4} x - \cos^ {4} x = (\sin^ {2} x + \cos^ {2} x) (\sin^ {2} x - \cos^ {2} x) =$$

Using the basic relationship between the sine and the cosine $\sin^2 x + \cos^2 x = 1$ , we get:

$$= (1) \cdot (\sin^ {2} x - \cos^ {2} x) = \sin^ {2} x - \cos^ {2} x = \sin^ {2} x - \cos^ {2} x + 1 - 1$$

We add and subtract 1(one):

$$= \sin^ {2} x - \cos^ {2} x + 1 - 1$$

Using the basic relationship between the sine and the cosine, we have:

$$= \sin^ {2} x - \cos^ {2} x + \sin^ {2} x + \cos^ {2} x - 1 =$$

And simplified:

$$= 2 \sin^ {2} x - 1$$

And this is the same as the right part of the expression.

Solution: expressions are identical.