Understanding the Trigonometric Expression

The problem asks us to evaluate the expression \({\left( {\frac{{sin27^\circ }}{{cos63^\circ }} - \frac{{cos27^\circ }}{{sin63^\circ }}} \right)^2}\). This expression involves trigonometric ratios of angles that seem different (27° and 63°), but they are related through the concept of complementary angles.

Applying Complementary Angle Identities

Two angles are called complementary if their sum is 90°. In this case, 27° + 63° = 90°, so 27° and 63° are complementary angles.

We can use the following complementary angle identities:

• \(\sin(90^\circ - \theta) = \cos \theta\)
• \(\cos(90^\circ - \theta) = \sin \theta\)

Using these identities, we can relate the trigonometric ratios of 63° to those of 27°:

• \(\cos63^\circ = \cos(90^\circ - 27^\circ) = \sin27^\circ\)
• \(\sin63^\circ = \sin(90^\circ - 27^\circ) = \cos27^\circ\)

Simplifying the Expression

Now, substitute these equivalent expressions back into the original equation:

\({\left( {\frac{{sin27^\circ }}{{cos63^\circ }} - \frac{{cos27^\circ }}{{sin63^\circ }}} \right)^2}\)

Substitute \(\cos63^\circ = \sin27^\circ\) and \(\sin63^\circ = \cos27^\circ\):

\({\left( {\frac{{sin27^\circ }}{{sin27^\circ }} - \frac{{cos27^\circ }}{{cos27^\circ }}} \right)^2}\)

Assuming \(\sin27^\circ \ne 0\) and \(\cos27^\circ \ne 0\) (which is true for 27°), the fractions simplify:

• \(\frac{{sin27^\circ }}{{sin27^\circ }} = 1\)
• \(\frac{{cos27^\circ }}{{cos27^\circ }} = 1\)

So the expression becomes:

\({\left( {1 - 1} \right)^2}\)

Perform the subtraction inside the parenthesis:

\({\left( 0 \right)^2}\)

Finally, calculate the square:

\({0^2} = 0\)

Step-by-Step Calculation

Let's break down the solution process:

1. Identify the angles in the expression: 27° and 63°.
2. Check if they are complementary: 27° + 63° = 90°. Yes, they are.
3. Use complementary angle identities: \(\cos63^\circ = \sin27^\circ\) and \(\sin63^\circ = \cos27^\circ\).
4. Substitute these into the expression: \({\left( {\frac{{sin27^\circ }}{{\sin27^\circ }} - \frac{{cos27^\circ }}{{\cos27^\circ }}} \right)^2}\).
5. Simplify the fractions: \({\left( {1 - 1} \right)^2}\).
6. Calculate the value: \({\left( 0 \right)^2} = 0\).

The final value of the given expression is 0.

Revision Table: Key Trigonometry Concepts

Additional Information: Complementary Angles in Practice

Understanding complementary angles is very useful in trigonometry. It allows us to simplify expressions involving angles like 10° and 80°, 45° and 45°, etc. For instance, if you need to find \(\tan 80^\circ\) but only know values for angles up to 45°, you can use the identity \(\tan(90^\circ - \theta) = \cot \theta\). So, \(\tan 80^\circ = \tan(90^\circ - 10^\circ) = \cot 10^\circ\).

The co-function identities (like \(\sin \leftrightarrow \cos\), \(\tan \leftrightarrow \cot\), \(\sec \leftrightarrow \csc\)) stem directly from the concept of complementary angles in a right-angled triangle. If one acute angle is \(\theta\), the other acute angle is \(90^\circ - \theta\).