Proving the Trigonometric Identity: cos^4 t - sin^4 t cos^2 t - sin^2 t

In this article, we will walk through a detailed proof of the trigonometric identity: cos^4 t - sin^4 t cos^2 t - sin^2 t. This identity involves the basic properties of trigonometric functions and can be a useful tool in various mathematical and engineering scenarios.

Introduction to the Identity

First, let's recall a fundamental algebraic identity:

a^2 - b^2 (a b)(a - b)

By applying this identity to the left-hand side (LHS) of the given trigonometric expression, we can simplify the equation more easily.

Step-by-Step Proof

Let's begin with the LHS of the equation:

cos^4 t - sin^4 t

We can factor this expression as follows:

cos^4 t - sin^4 t (cos^2 t sin^2 t)(cos^2 t - sin^2 t)

Now, we know from the Pythagorean identity that:

cos^2 t sin^2 t 1

Substituting this into the equation, we get:

cos^4 t - sin^4 t 1 * (cos^2 t - sin^2 t)

Which simplifies to:

cos^4 t - sin^4 t cos^2 t - sin^2 t

Alternative Proof

We can also prove the identity by using another approach:

LHS cos^4 t - sin^4 t

Using the difference of squares formula, we can rewrite this as:

cos^4 t - sin^4 t (cos^2 t sin^2 t)(cos^2 t - sin^2 t)

Again, using the Pythagorean identity, we know that:

cos^2 t sin^2 t 1

Substituting this, we get:

cos^4 t - sin^4 t 1 * (cos^2 t - sin^2 t)

Which simplifies to:

cos^4 t - sin^4 t cos^2 t - sin^2 t

Generalization to a^2 - b^2

More generally, we can express:

a^4 - b^4 (a^2 b^2)(a^2 - b^2)

And, if we let:

a cos t and b sin t

We get:

cos^4 t - sin^4 t (cos^2 t sin^2 t)(cos^2 t - sin^2 t)

Again, using the Pythagorean identity, we know:

cos^2 t sin^2 t 1

Thus, the equation simplifies to:

cos^4 t - sin^4 t 1 * (cos^2 t - sin^2 t)

Which results in:

cos^4 t - sin^4 t cos^2 t - sin^2 t

Conclusion

In conclusion, we have shown that the trigonometric identity: cos^4 t - sin^4 t cos^2 t - sin^2 t holds true. This identity can be extremely useful in simplifying complex trigonometric expressions and solving related problems. Whether you use factoring techniques or the Pythagorean identity, the proof is straightforward and elegant.

Keywords: trigonometric identity, proof, cosine and sine