Understanding Trigonometric Identities: Finding tan(\(\theta\)/2)

The question asks us to find an expression that is equivalent to \(\tan \frac{\theta}{2}\) from the given options. This involves understanding and applying fundamental trigonometric identities, specifically the half-angle identities.

Trigonometric identities are equations that are true for all values of the variables for which both sides of the equation are defined. The half-angle identities relate the trigonometric functions of an angle to the trigonometric functions of half that angle.

Deriving the Formula for tan(\(\theta\)/2)

We can derive one form of the identity for \(\tan \frac{\theta}{2}\) using the double-angle formulas for sine and cosine:

• The double angle formula for sine is \(\sin (2x) = 2 \sin x \cos x\). If we let \(2x = \theta\), then \(x = \frac{\theta}{2}\). So, \(\sin \theta = 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}\).
• There are several double angle formulas for cosine. One useful one is \(\cos (2x) = 2 \cos^2 x - 1\). Again, letting \(2x = \theta\), we get \(\cos \theta = 2 \cos^2 \frac{\theta}{2} - 1\). Rearranging this, we get \(1 + \cos \theta = 2 \cos^2 \frac{\theta}{2}\).

Now consider the expression \(\frac{\sin \theta}{1 + \cos \theta}\) from the options. Substitute the expressions we derived:

\[ \frac{\sin \theta}{1 + \cos \theta} = \frac{2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}}{2 \cos^2 \frac{\theta}{2}} \]

We can cancel out the common factors of 2 and \(\cos \frac{\theta}{2}\) from the numerator and the denominator, provided \(\cos \frac{\theta}{2} \neq 0\):

\[ \frac{2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}}{2 \cos^2 \frac{\theta}{2}} = \frac{\sin \frac{\theta}{2}}{\cos \frac{\theta}{2}} \]

By the definition of the tangent function, \(\tan x = \frac{\sin x}{\cos x}\). Therefore, \(\frac{\sin \frac{\theta}{2}}{\cos \frac{\theta}{2}} = \tan \frac{\theta}{2}\).

So, we have shown that \(\tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta}\).

Comparing with the Options

Let's compare our result with the given options:

• Option 1: \(\frac{\sin \theta}{1+\cos \theta}\) - This matches our derived identity.
• Option 2: \(\frac{\cos \theta}{1-\cos \theta}\) - This is not a standard identity for \(\tan \frac{\theta}{2}\).
• Option 3: \(\frac{\cos \theta}{1-\sin \theta}\) - This is not a standard identity for \(\tan \frac{\theta}{2}\).
• Option 4: \(\frac{\sin \theta}{1-\cos \theta}\) - This expression is actually equal to \(\cot \frac{\theta}{2}\), which is the reciprocal of \(\tan \frac{\theta}{2}\).

Based on the derivation, the expression equivalent to \(\tan \frac{\theta}{2}\) is \(\frac{\sin \theta}{1 + \cos \theta}\).

Revision Table: Key Trigonometric Half-Angle Identities

Additional Information on Half-Angle Formulas

The half-angle formulas are very useful in trigonometry for simplifying expressions or finding exact values of trigonometric functions for angles that are half of common angles (like 15 degrees, which is half of 30 degrees). They are all derived from the double-angle or power-reducing identities.

For example, the identity \(\tan \frac{\theta}{2} = \frac{1 - \cos \theta}{\sin \theta}\) can be derived similarly:

\[ \frac{1 - \cos \theta}{\sin \theta} = \frac{1 - (1 - 2 \sin^2 \frac{\theta}{2})}{2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}} = \frac{2 \sin^2 \frac{\theta}{2}}{2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}} = \frac{\sin \frac{\theta}{2}}{\cos \frac{\theta}{2}} = \tan \frac{\theta}{2} \]

These different forms of the half-angle identities for tangent are often interchangeable, but one form might be more convenient than others depending on the specific problem.