Simplifying Trigonometric Expressions Using Identities

We are asked to simplify the given trigonometric expression:

\(\frac{{{{\sin }^3}{\rm{A}} + \sin 3{\rm{\;A}}}}{{\sin {\rm{A}}}} + \frac{{{{\cos }^3}{\rm{A}} - \cos 3{\rm{\;A}}}}{{\cos {\rm{A}}}}\)

To simplify this expression, we will use the triple angle identities for sine and cosine:

• \(\sin 3{\rm{A}} = 3\sin {\rm{A}} - 4{\sin ^3}{\rm{A}}\)
• \(\cos 3{\rm{A}} = 4{\cos ^3}{\rm{A}} - 3\cos {\rm{A}}\)

Let's simplify the first term of the expression:

Simplifying the First Term: \(\frac{{{{\sin }^3}{\rm{A}} + \sin 3{\rm{\;A}}}}{{\sin {\rm{A}}}}\)

Substitute the identity for \(\sin 3{\rm{A}}\) into the numerator:

\(\sin^3 {\rm{A}} + \sin 3{\rm{A}} = \sin^3 {\rm{A}} + (3\sin {\rm{A}} - 4\sin^3 {\rm{A}})\)

\(= 3\sin {\rm{A}} + \sin^3 {\rm{A}} - 4\sin^3 {\rm{A}}\)

\(= 3\sin {\rm{A}} - 3\sin^3 {\rm{A}}\)

\(= 3\sin {\rm{A}} (1 - \sin^2 {\rm{A}})\)

Now, substitute this back into the first term expression (assuming \(\sin {\rm{A}} \neq 0\)):

\(\frac{{3\sin {\rm{A}} (1 - \sin^2 {\rm{A}})}}{{\sin {\rm{A}}}}\)

\(= 3 (1 - \sin^2 {\rm{A}})\)

Using the Pythagorean identity \(\sin^2 {\rm{A}} + \cos^2 {\rm{A}} = 1\), we know that \(1 - \sin^2 {\rm{A}} = \cos^2 {\rm{A}}\).

So, the first term simplifies to:

\(= 3 \cos^2 {\rm{A}}\)

Simplifying the Second Term: \(\frac{{{{\cos }^3}{\rm{A}} - \cos 3{\rm{\;A}}}}{{\cos {\rm{A}}}}\)

Substitute the identity for \(\cos 3{\rm{A}}\) into the numerator:

\(\cos^3 {\rm{A}} - \cos 3{\rm{A}} = \cos^3 {\rm{A}} - (4\cos^3 {\rm{A}} - 3\cos {\rm{A}})\)

\(= \cos^3 {\rm{A}} - 4\cos^3 {\rm{A}} + 3\cos {\rm{A}}\)

\(= 3\cos {\rm{A}} - 3\cos^3 {\rm{A}}\)

\(= 3\cos {\rm{A}} (1 - \cos^2 {\rm{A}})\)

Now, substitute this back into the second term expression (assuming \(\cos {\rm{A}} \neq 0\)):

\(\frac{{3\cos {\rm{A}} (1 - \cos^2 {\rm{A}})}}{{\cos {\rm{A}}}}\)

\(= 3 (1 - \cos^2 {\rm{A}})\)

Using the Pythagorean identity \(\sin^2 {\rm{A}} + \cos^2 {\rm{A}} = 1\), we know that \(1 - \cos^2 {\rm{A}} = \sin^2 {\rm{A}}\).

So, the second term simplifies to:

\(= 3 \sin^2 {\rm{A}}\)

Adding the Simplified Terms:

The original expression is the sum of the simplified first and second terms:

\(\frac{{{{\sin }^3}{\rm{A}} + \sin 3{\rm{\;A}}}}{{\sin {\rm{A}}}} + \frac{{{{\cos }^3}{\rm{A}} - \cos 3{\rm{\;A}}}}{{\cos {\rm{A}}}} = 3 \cos^2 {\rm{A}} + 3 \sin^2 {\rm{A}}\)

\(= 3 (\cos^2 {\rm{A}} + \sin^2 {\rm{A}})\)

Using the Pythagorean identity \(\sin^2 {\rm{A}} + \cos^2 {\rm{A}} = 1\):

\(= 3 (1)\)

\(= 3\)

Thus, the simplified expression is 3.

Revision Table: Key Trigonometric Identities

Additional Information: Understanding Trigonometric Simplification

Trigonometric simplification involves using various trigonometric identities to reduce a complex expression into a simpler form. This is a fundamental skill in trigonometry and is often required when solving equations, evaluating integrals, or analyzing periodic functions.

The identities used here, specifically the triple angle identities for \(\sin 3{\rm{A}}\) and \(\cos 3{\rm{A}}\), are derived from angle addition formulas. For example, \(\sin 3{\rm{A}} = \sin (2{\rm{A}} + {\rm{A}})\) can be expanded using the \(\sin(X+Y)\) formula and then further simplified using double angle identities (\(\sin 2{\rm{A}}\) and \(\cos 2{\rm{A}}\)). Mastering these core identities is crucial for simplifying more complex trigonometric expressions.

Always remember to note any restrictions on the variable \({\rm{A}}\), such as \(\sin {\rm{A}} \neq 0\) and \(\cos {\rm{A}} \neq 0\), which ensure the denominators are non-zero when simplifying by dividing. However, the final result '3' is a constant, implying the expression evaluates to 3 for all valid values of A where the original expression is defined.