Understanding the Tan 54° Expression

The question asks us to express \(\tan 54^\circ\) in a different form using trigonometric functions of other angles, as provided in the options. The options involve angles like \(9^\circ\) and \(36^\circ\). We need to find a relationship between \(54^\circ\) and these angles that can simplify the expression.

Relating Angles: 54° and 9°

Let's look at the angle \(54^\circ\). We can express \(54^\circ\) as a sum or difference of standard angles or angles appearing in the options. A useful observation is that \(54^\circ = 45^\circ + 9^\circ\). This is helpful because \(45^\circ\) is a standard angle whose trigonometric values are known, and \(9^\circ\) appears in some of the options.

Applying the Tangent Addition Formula

We can use the tangent addition formula, which states:

Let \(A = 45^\circ\) and \(B = 9^\circ\). Substituting these values into the formula:

We know that the value of \(\tan 45^\circ\) is 1. Substituting this value:

Converting Tangent to Sine and Cosine

The options are given in terms of sine and cosine of \(9^\circ\). We know that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Let's substitute \(\tan 9^\circ = \frac{\sin 9^\circ}{\cos 9^\circ}\) into our expression:

To simplify this complex fraction, we can multiply both the numerator and the denominator by \(\cos 9^\circ\):

Comparing with the Options

Let's compare our derived expression with the given options:

Option 1: \(\frac{{\sin 9^\circ + \cos 9^\circ }}{{sin9^\circ - \cos 9^\circ }}\)

Option 2: \(\frac{{\sin 9^\circ - \;\cos 9^\circ }}{{sin9^\circ + \cos 9^\circ }}\)

Option 3: \(\frac{{\cos 9^\circ + \sin 9^\circ }}{{\cos 9^\circ - \sin 9^\circ }}\)

Option 4: \(\frac{{\sin 36^\circ }}{{\cos 36^\circ }}\)

Our derived expression \(\frac{\cos 9^\circ + \sin 9^\circ}{\cos 9^\circ - \sin 9^\circ}\) exactly matches Option 3.

Let's quickly analyze Option 4. \(\frac{\sin 36^\circ}{\cos 36^\circ} = \tan 36^\circ\). Since \(54^\circ \neq 36^\circ\), \(\tan 54^\circ \neq \tan 36^\circ\). So Option 4 is incorrect.

Let's examine Option 1 and 2 more closely by dividing numerator and denominator by \(\cos 9^\circ\):

Option 1: \(\frac{\frac{\sin 9^\circ}{\cos 9^\circ} + \frac{\cos 9^\circ}{\cos 9^\circ}}{\frac{\sin 9^\circ}{\cos 9^\circ} - \frac{\cos 9^\circ}{\cos 9^\circ}} = \frac{\tan 9^\circ + 1}{\tan 9^\circ - 1} = \frac{1 + \tan 9^\circ}{-(1 - \tan 9^\circ)} = - \frac{1 + \tan 9^\circ}{1 - \tan 9^\circ} = -\tan 54^\circ\). This is incorrect.

Option 2: \(\frac{\frac{\sin 9^\circ}{\cos 9^\circ} - \frac{\cos 9^\circ}{\cos 9^\circ}}{\frac{\sin 9^\circ}{\cos 9^\circ} + \frac{\cos 9^\circ}{\cos 9^\circ}} = \frac{\tan 9^\circ - 1}{\tan 9^\circ + 1}\). Using the identity \(\tan(45^\circ - \theta) = \frac{\tan 45^\circ - \tan \theta}{1 + \tan 45^\circ \tan \theta} = \frac{1 - \tan \theta}{1 + \tan \theta}\), we see \(\frac{\tan 9^\circ - 1}{\tan 9^\circ + 1} = - \frac{1 - \tan 9^\circ}{1 + \tan 9^\circ} = -\tan(45^\circ - 9^\circ) = -\tan 36^\circ\). Alternatively, \(\frac{\tan 9^\circ - 1}{\tan 9^\circ + 1} = \frac{\frac{1+\tan 9^\circ}{1-\tan 9^\circ} - 2}{1 + \frac{1+\tan 9^\circ}{1-\tan 9^\circ}}\). This path is more complex. Let's use the \(\tan 54^\circ\) result. We have \(\frac{1+\tan 9^\circ}{1-\tan 9^\circ}\). Option 2 is the reciprocal of Option 1, so it is \(\frac{1}{-\tan 54^\circ} = -\cot 54^\circ = -\tan(90^\circ - 54^\circ) = -\tan 36^\circ\). This is also incorrect.

Thus, the only option that matches our derived expression for \(\tan 54^\circ\) is Option 3.

Conclusion

By expressing \(\tan 54^\circ\) as \(\tan(45^\circ + 9^\circ)\) and using the tangent addition formula, we successfully transformed the expression into the form given in Option 3.

Revision Table: Key Trigonometry Concepts

Additional Information: Exploring Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables for which both sides of the equation are defined. They are crucial for simplifying expressions and solving trigonometric equations.

Besides the addition formulas used here, there are many other important identities:

Pythagorean Identities: \(\sin^2 \theta + \cos^2 \theta = 1\), \(1 + \tan^2 \theta = \sec^2 \theta\), \(1 + \cot^2 \theta = \csc^2 \theta\).

Double Angle Formulas: \(\sin 2\theta = 2 \sin \theta \cos \theta\), \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\), \(\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}\).

Complementary Angle Identities: For angles \(A\) and \(B\) such that \(A+B = 90^\circ\), \(\sin A = \cos B\), \(\tan A = \cot B\), \(\sec A = \csc B\). In this problem, \(54^\circ + 36^\circ = 90^\circ\), so \(\tan 54^\circ = \cot 36^\circ\). Also, \(\frac{\sin 36^\circ}{\cos 36^\circ} = \tan 36^\circ\). Since \(\cot 36^\circ = \frac{1}{\tan 36^\circ}\), \(\tan 54^\circ = \frac{1}{\tan 36^\circ}\). This confirms why Option 4 was incorrect, as it was just \(\tan 36^\circ\).

Mastering these identities helps in manipulating trigonometric expressions efficiently.