Evaluating the Trigonometric Expression

We need to evaluate the given trigonometric expression:

\[ \frac{{1 - \tan 2^\circ \cot 62^\circ }}{{\tan 152^\circ - \cot 88^\circ }} \]

Let's simplify the numerator and the denominator separately using trigonometric identities and angle properties.

Simplifying the Denominator

The denominator is \( \tan 152^\circ - \cot 88^\circ \). We can rewrite the angles using standard transformations:

• For \( \tan 152^\circ \): We know that \( 152^\circ = 180^\circ - 28^\circ \). The identity for \( \tan(180^\circ - \theta) \) is \( -\tan \theta \).
• So, \( \tan 152^\circ = \tan(180^\circ - 28^\circ) = -\tan 28^\circ \).
• For \( \cot 88^\circ \): We know that \( 88^\circ = 90^\circ - 2^\circ \). The identity for \( \cot(90^\circ - \theta) \) is \( \tan \theta \).
• So, \( \cot 88^\circ = \cot(90^\circ - 2^\circ) = \tan 2^\circ \).

Substituting these back into the denominator:

\[ \tan 152^\circ - \cot 88^\circ = (-\tan 28^\circ) - (\tan 2^\circ) = -(\tan 28^\circ + \tan 2^\circ) \]

Simplifying the Numerator

The numerator is \( 1 - \tan 2^\circ \cot 62^\circ \). We can rewrite \( \cot 62^\circ \):

• For \( \cot 62^\circ \): We know that \( 62^\circ = 90^\circ - 28^\circ \). The identity for \( \cot(90^\circ - \theta) \) is \( \tan \theta \).
• So, \( \cot 62^\circ = \cot(90^\circ - 28^\circ) = \tan 28^\circ \).

Substituting this back into the numerator:

\[ 1 - \tan 2^\circ \cot 62^\circ = 1 - \tan 2^\circ \tan 28^\circ \]

Evaluating the Full Expression

Now substitute the simplified numerator and denominator back into the original expression:

\[ \frac{{1 - \tan 2^\circ \cot 62^\circ }}{{\tan 152^\circ - \cot 88^\circ }} = \frac{{1 - \tan 2^\circ \tan 28^\circ}}{{-(\tan 28^\circ + \tan 2^\circ)}} \]

We can rewrite this as:

\[ - \frac{{1 - \tan 2^\circ \tan 28^\circ}}{{\tan 2^\circ + \tan 28^\circ}} \]

Recall the tangent addition formula: \( \tan(A + B) = \frac{{\tan A + \tan B}}{{1 - \tan A \tan B}} \). Taking the reciprocal, we get \( \cot(A + B) = \frac{{1 - \tan A \tan B}}{{\tan A + \tan B}} \).

Comparing the expression \( \frac{{1 - \tan 2^\circ \tan 28^\circ}}{{\tan 2^\circ + \tan 28^\circ}} \) with the \( \cot(A + B) \) formula, we can set \( A = 2^\circ \) and \( B = 28^\circ \).

So, \( \frac{{1 - \tan 2^\circ \tan 28^\circ}}{{\tan 2^\circ + \tan 28^\circ}} = \cot(2^\circ + 28^\circ) = \cot(30^\circ) \).

The value of \( \cot(30^\circ) \) is \( \sqrt{3} \).

Substituting this back into our full expression:

\[ - \cot(30^\circ) = -\sqrt{3} \]

Thus, the value of the given expression is \( -\sqrt{3} \).

Revision Table: Key Trigonometric Identities Used

Additional Information: Angle Transformations and Exact Values

Understanding angle transformations (like \( 90^\circ \pm \theta \), \( 180^\circ \pm \theta \), etc.) is crucial for simplifying trigonometric expressions. These transformations allow us to relate trigonometric functions of angles outside the first quadrant to functions of acute angles.

For example:

• \( \tan(90^\circ - \theta) = \cot \theta \)
• \( \sin(180^\circ - \theta) = \sin \theta \)
• \( \cos(180^\circ - \theta) = -\cos \theta \)
• \( \sec(90^\circ + \theta) = -\csc \theta \)

Knowing the exact values of trigonometric functions for standard angles like \( 0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ \) (and their multiples) is also essential. These values often appear in simplified expressions.

By combining angle transformations and fundamental identities, complex expressions can often be reduced to simple values.