Want to know how to calculate trig identities (equations that have trig functions)? In this post, I’ll explain how to use a well-known formula (difference of squares) along with some well-known trig identities (Pythagorean identity and angle sum identities) to determine a trig identity. Knowing how to make calculations such as these can be very useful.
Let’s look at this equation:
$$ \cos^4(x) \, – \, \sin^4(x) = \cos(2x) $$
Note that
$$ \cos^4(x) = \cos^2(x) \cos^2(x) = (\cos^2(x))^2 $$
and
$$ \sin^4(x) = \sin^2(x) \sin^2(x) = (\sin^2(x))^2.$$
So we can re-write the LHS as follows:
$$ \cos^4(x) \, – \, \sin^4(x) = (\cos^2(x))^2 \, – \, (\sin^2(x))^2 $$
Now, we use Difference of Squares to calculate:
$$ (\cos^2(x))^2 \, – \, (\sin^2(x))^2 =$$
$$(\cos^2(x) + \sin^2(x)) (\cos^2(x) \, – \, \sin^2(x))$$
An important trig identity to remember is the following (called the Pythagorean Identity):
$$ \cos^2(x) + \sin^2(x) = 1 $$
So,
$$ (\cos^2(x) + \sin^2(x)) (\cos^2(x) \, – \, \sin^2(x))$$
$$ = \cos^2(x) \, – \, \sin^2(x)$$
Now, we use the following angle sum identity for cosine:
$$ \cos(a + b) = \cos(a) \cos(b) \, – \, \sin(a) \sin(b) $$
Letting \(a = x\) and letting \( b = x\), we obtain:
$$ \cos(x + x) = \cos(x) \cos(x) \, – \, \sin(x) \sin(x) $$
So,
$$ \cos(2x) = \cos^2(x) \, – \, \sin^2(x) $$
Putting all this together, we obtain
$$ \cos^4(x) \, – \, \sin^4(x) = \cos(2x) $$
