Trigonometry Identity Simplification

The question asks us to simplify the given trigonometric expression and find which standard form it is equal to.

The expression provided is: \(\frac{{2\tan \theta }}{{1 + {{\tan }^2}\theta }}\)

To simplify this expression, we can use fundamental trigonometric identities. One key identity relates \(1 + \tan^2 \theta\) to another trigonometric function.

Using Trigonometric Identities

We know the Pythagorean identity involving tangent and secant:

• \(1 + {{\tan }^2}\theta = {{\sec }^2}\theta\)

Let's substitute this into the denominator of the given expression:

\(\frac{{2\tan \theta }}{{1 + {{\tan }^2}\theta }} = \frac{{2\tan \theta }}{{{{\sec }^2}\theta }}\)

Now, we can express \(\tan \theta\) and \(\sec \theta\) in terms of \(\sin \theta\) and \(\cos \theta\):

• \(\tan \theta = \frac{{\sin \theta }}{{\cos \theta }}\)
• \(\sec \theta = \frac{1}{{\cos \theta }}\)
• Therefore, \({{\sec }^2}\theta = {\left( {\frac{1}{{\cos \theta }}} \right)^2} = \frac{1}{{{{\cos }^2}\theta }}\)

Substitute these into the simplified expression:

\(\frac{{2\tan \theta }}{{{{\sec }^2}\theta }} = \frac{{2\left( {\frac{{\sin \theta }}{{\cos \theta }}} \right)}}{{\frac{1}{{{{\cos }^2}\theta }}}}\)

To simplify this complex fraction, we multiply the numerator by the reciprocal of the denominator:

\(\frac{{2\left( {\frac{{\sin \theta }}{{\cos \theta }}} \right)}}{{\frac{1}{{{{\cos }^2}\theta }}}} = 2\left( {\frac{{\sin \theta }}{{\cos \theta }}} \right) \times \left( {{{\cos }^2}\theta } \right)\)

Now, we can cancel one factor of \(\cos \theta\) from the numerator and the denominator:

\(2\frac{{\sin \theta }}{{\cancel{{\cos \theta }}}} \times {{\cos }^{\cancel{2}}}\theta = 2\sin \theta \cos \theta\)

The resulting expression is \(2\sin \theta \cos \theta\). This is a well-known double angle identity.

• \(\sin 2\theta = 2\sin \theta \cos \theta\)

Therefore, the given expression \(\frac{{2\tan \theta }}{{1 + {{\tan }^2}\theta }}\) simplifies to \(\sin 2\theta\).

Comparing with Options

Let's check the given options:

1. \(\cos 2\theta\)
2. \(\tan 2\theta\)
3. \(\sin 2\theta\)
4. \(\csc 2\theta\)

Our simplified expression matches option 3, which is \(\sin 2\theta\).

Thus, the expression \(\frac{{2\tan \theta }}{{1 + {{\tan }^2}\theta }}\) is equal to \(\sin 2\theta\).

Revision Table: Key Trigonometry Identities

Additional Information on Trigonometric Identities

Trigonometric identities are equations that are true for all possible values of the variables involved. They are fundamental tools in trigonometry for simplifying expressions, solving equations, and proving other relationships.

The identity we used, \(\frac{{2\tan \theta }}{{1 + {{\tan }^2}\theta }} = \sin 2\theta\), is actually one form of the double angle identity for sine, specifically expressed in terms of \(\tan \theta\). This form is particularly useful when dealing with expressions involving \(\tan \theta\).

It's important to recognize these different forms of identities as they can greatly simplify trigonometric problems. For instance, the double angle formula for cosine also has several forms involving \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\).

Mastering these identities requires practice. Understanding the relationships between different trigonometric functions (\(\sin\), \(\cos\), \(\tan\), \(\sec\), \(\csc\), \(\cot\)) and how they relate through fundamental identities (like Pythagorean and reciprocal identities) is key to simplifying more complex expressions.