91Ó°ÊÓ

The double angle identities are powerful tools in trigonometry that help us simplify expressions where the angle is doubled. The most common double angle identities are for sine and cosine:

\[\sin 2x = 2 \sin x \cos x \]
\[\cos 2x = 2 \cos^2 x - 1 \]
These identities allow us to express trigonometric functions of double angles in terms of single angles. This can simplify the verification of trigonometric identities and make it easier to solve trigonometric equations.

In this exercise, we used the double angle identities to rewrite the given expression by replacing \(\sin 4x\) and \(\cos 4x\) with their respective double angle identities. This step is crucial because it helps us transform the expression into a form that is more manageable for further simplification.

The tangent addition formula is instrumental in trigonometry when you need to add two angles together. The formula is given as:

\[\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
This formula simplifies the task of finding the tangent of the sum of two angles by using their individual tangents. It is particularly useful for verifying and proving trigonometric identities involving tangent.

In the exercise solution, after breaking down the problem using the double angle identities and simplifying, we arrive at a point where the problem requires the use of the tangent addition formula. By recognizing \(a = x\) and \(b = 2x\), we can substitute into the formula, ultimately leading to \(\tan 3x\), which was the target expression we needed to verify.

Simplifying trigonometric expressions is often the core of solving trigonometric identities. It involves using known identities like double angle identities and addition formulas to rewrite complex expressions more simply. Steps include:

• Combining like terms, which helps in organizing terms and making the expression less complex.
• Factoring out common terms, which simplifies denominators and numerators by reducing them to more straightforward components.
• Applying fundamental trigonometric formulas, such as the sine, cosine, or tangent of an angle, to reduce the expression.

These techniques were pivotal in resolving the tangent identity in the exercise. By successively applying these simplification strategies, including the rearrangement of terms and appropriate application of trigonometric identities, we were able to bridge the original problem to its simplified solution with \(\tan 3x\) as the final result.