Darryl B. answered • 04/25/19
IVY Quantum Theorist for Online Math and Physics Tutorials
Hi Nick,
We can use a trick of algebraic factoring to simplify this expression. Recall that
a2 - b2 = (a + b)(a - b).
In this case, let's simplify our expression in terms of the above relation:
(sin4x - cos4x) = (sin2x)2 - (cos2x)2
= [(sin2x + cos2x)(sin2x - cos2x)].
The expression in bold is the Pythagorean Identity for trig functions: it is equal to 1. Using the same identity, we can also replace one of the squared trig function, we have
(sin4x - cos4x) = (sin2x)2 - (cos2x)2
= [(sin2x + cos2x)(sin2x - cos2x)].
= [(1)(sin2x - cos2x)]
= {(1)[(1 - cos2x) - cos2x]
= (1)(1 - 2cos2x)
= 1 - 2cos2x.
This is the simplified version of the expression. Let me know if this makes sense, and/or if you have any question. I hope this helps!
Darryl B.
04/25/19
Victoria V.
One more addition, it can be simplified one step further to a single trig function... 1 - 2(cosx)^2 does EQUAL -cos(2x), which is probably considered a "more simplified" answer, taking this last step.04/25/19