Simplifying Trigonometric Expressions: 1 + tan 15° cot 75°

Let's simplify the given trigonometric expression: \(1 + \tan 15^\circ \cot 75^\circ\).

To simplify this expression, we can use trigonometric identities. We notice that the angles are \(15^\circ\) and \(75^\circ\). These are complementary angles because \(15^\circ + 75^\circ = 90^\circ\).

Using Complementary Angle Identities

A key identity for complementary angles is that \( \cot (90^\circ - \theta) = \tan \theta \) and \( \tan (90^\circ - \theta) = \cot \theta \).

We can rewrite \( \cot 75^\circ \) using the complementary angle identity:

\( \cot 75^\circ = \cot (90^\circ - 15^\circ) \)

Using the identity \( \cot (90^\circ - \theta) = \tan \theta \) with \( \theta = 15^\circ \):

\( \cot (90^\circ - 15^\circ) = \tan 15^\circ \)

So, \( \cot 75^\circ = \tan 15^\circ \).

Substituting and Simplifying the Expression

Now substitute this back into the original expression:

\( 1 + \tan 15^\circ \cot 75^\circ \)

Replace \( \cot 75^\circ \) with \( \tan 15^\circ \):

\( 1 + \tan 15^\circ \cdot \tan 15^\circ \)

This simplifies to:

\( 1 + \tan^2 15^\circ \)

Using Pythagorean Identities

Now we can use a Pythagorean identity. The Pythagorean identities relate squares of trigonometric functions. One such identity is:

\( 1 + \tan^2 \theta = \sec^2 \theta \)

Applying this identity with \( \theta = 15^\circ \):

\( 1 + \tan^2 15^\circ = \sec^2 15^\circ \)

Conclusion

Therefore, the expression \( 1 + \tan 15^\circ \cot 75^\circ \) simplifies to \( \sec^2 15^\circ \).

Summary of Steps

• Start with the given expression \(1 + \tan 15^\circ \cot 75^\circ\).
• Recognize that \(15^\circ\) and \(75^\circ\) are complementary angles.
• Use the complementary angle identity \( \cot (90^\circ - \theta) = \tan \theta \) to write \( \cot 75^\circ \) as \( \tan 15^\circ \).
• Substitute \( \tan 15^\circ \) for \( \cot 75^\circ \) in the expression, getting \( 1 + \tan 15^\circ \cdot \tan 15^\circ = 1 + \tan^2 15^\circ \).
• Use the Pythagorean identity \( 1 + \tan^2 \theta = \sec^2 \theta \) to simplify \( 1 + \tan^2 15^\circ \) to \( \sec^2 15^\circ \).

Revision Table: Key Trigonometric Identities Used

Additional Information: Understanding Complementary Angles and Identities

Complementary angles are two angles that add up to \(90^\circ\). Trigonometric functions of complementary angles are related in specific ways. For example:

• \( \sin (90^\circ - \theta) = \cos \theta \)
• \( \cos (90^\circ - \theta) = \sin \theta \)
• \( \tan (90^\circ - \theta) = \cot \theta \)
• \( \cot (90^\circ - \theta) = \tan \theta \)
• \( \sec (90^\circ - \theta) = \cosec \theta \)
• \( \cosec (90^\circ - \theta) = \sec \theta \)

These identities are useful for simplifying expressions involving angles like \(15^\circ\) and \(75^\circ\), \(30^\circ\) and \(60^\circ\), or \(1^\circ\) and \(89^\circ\).

Pythagorean identities come from the Pythagorean theorem applied to a right-angled triangle or the unit circle. The three main Pythagorean identities are:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)
• \( 1 + \tan^2 \theta = \sec^2 \theta \)
• \( 1 + \cot^2 \theta = \cosec^2 \theta \)

These identities are fundamental in trigonometry and are frequently used to simplify expressions or prove other identities.