• We will analyze trigonometric identities numerically and graphically.
• We will discuss techniques used to manipulate and simplify expressions in order to prove trigonometric identities algebraically.

Recall: A trigonometric identity is an equation formed by the equivalence of two trigonometric expressions.

The solution to the equation must be the set of all values of the variable, in which both expressions are defined.

Consider the trigonometric equation

Is this equation an identity?

Is this statement true for all values of \(x\)?

Solution

When \(x = \dfrac{\pi}{4}\),

Therefore, L.S. \(=\) R.S.

Consider the trigonometric equation

Is this equation an identity?

Is this statement true for all values of \(x\)?

Solution

When \(x = \dfrac{\pi}{2}\),

Therefore, L.S. \(=\) R.S.

This equality statement holds true for \(x = \dfrac{\pi}{4}\) and \(x = \dfrac{\pi}{2}\).

Consider the trigonometric equation

Is this equation an identity?

Is this statement true for all values of \(x\)?

Solution

Can we conclude that \(\sin^2(x) - \cos^2(x) = \sin(x) - \cos(x)\) is an identity?

If we graph the functions \(f(x) = \sin^2(x) - \cos^2(x)\) and \(g(x) = \sin(x) - \cos(x)\), using technology, we see that \(f(x) = g(x)\) for an infinite number of values of \(x\) due to the periodic nature of the two functions.

However, we cannot say \(f(x) = g(x)\) for all values of \(x\) within the domains of the two functions (in this case, for all real values of \(x\)). The graphs of \(f(x)\) and \(g(x)\) are not the same.

\(\sin^2(x) - \cos^2(x) = \sin(x) - \cos(x)\) is not an identity.

Consider the trigonometric equation

Is this equation an identity?

Is this statement true for all values of \(x\)?

Solution

We don't need to use this graphical approach to verify an equality statement is not an identity.

Finding one value of the variable that does not satisfy the equation is sufficient.

For example, when \(x = \dfrac{\pi}{3}\)

Therefore, L.S. \(\neq\) R.S.

Since \(x = \dfrac{\pi}{3}\) does not satisfy the equation, but is in the domain of the expression on each side of the equation, \(\sin^2(x) - \cos^2(x) = \sin(x) - \cos(x)\) is not an identity.

Consider the trigonometric equation

Is this equation an identity?

Is this statement true for all values of \(x\)?

Solution

We have shown numerically and graphically that \(\sin^2(x) - \cos^2(x) \neq \sin(x) - \cos(x)\) for all permissible values of \(x\).

Algebraically,

and

unless

Thus, \(\sin^2(x) - \cos^2(x) = \sin(x) - \cos(x)\) is not a trigonometric identity.