Simplifying Trigonometric Expressions

The problem asks us to simplify the trigonometric expression: \(\sin\theta \cdot \cos(90^\circ - \theta) + \cos\theta \cdot \sin(90^\circ - \theta)\).

To solve this, we can use fundamental trigonometric identities. There are a couple of ways to approach this, both relying on key identities.

Using Complementary Angle Identities

Recall the complementary angle identities:

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

Let's substitute these identities into the given expression:

Original expression: \(\sin\theta \cdot \cos(90^\circ - \theta) + \cos\theta \cdot \sin(90^\circ - \theta)\)

Substitute the identities:

\(\sin\theta \cdot (\sin\theta) + \cos\theta \cdot (\cos\theta)\)

This simplifies to:

\(\sin^2\theta + \cos^2\theta\)

Now, recall the Pythagorean identity:

• \(\sin^2\theta + \cos^2\theta = 1\)

Therefore, the expression simplifies to 1.

Using the Sum of Angles Identity

Another approach is to recognize the form of the expression. The given expression \(\sin\theta \cdot \cos(90^\circ - \theta) + \cos\theta \cdot \sin(90^\circ - \theta)\) matches the form of the sum of angles identity for sine:

• \(\sin(A+B) = \sin A \cos B + \cos A \sin B\)

In our expression, if we let \(A = \theta\) and \(B = 90^\circ - \theta\), the expression is exactly \(\sin A \cos B + \cos A \sin B\).

Using the identity, the expression is equal to \(\sin(A+B)\). Substitute the values of \(A\) and \(B\):

\(\sin(\theta + (90^\circ - \theta))\)

Simplify the angle inside the sine function:

\(\sin(\theta + 90^\circ - \theta) = \sin(90^\circ)\)

The value of \(\sin(90^\circ)\) is a standard trigonometric value:

• \(\sin(90^\circ) = 1\)

Thus, using the sum of angles identity also shows that the expression simplifies to 1.

Summary of Steps

We simplified the expression \(\sin\theta \cdot \cos(90^\circ - \theta) + \cos\theta \cdot \sin(90^\circ - \theta)\) using trigonometric identities.

• We used the complementary angle identities: \(\cos(90^\circ - \theta) = \sin\theta\) and \(\sin(90^\circ - \theta) = \cos\theta\).
• Substituting these gave us \(\sin\theta \cdot \sin\theta + \cos\theta \cdot \cos\theta = \sin^2\theta + \cos^2\theta\).
• Using the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\), we found the result is 1.

Alternatively:

• We recognized the pattern \(\sin A \cos B + \cos A \sin B\) with \(A = \theta\) and \(B = 90^\circ - \theta\).
• Using the sum of angles identity \(\sin(A+B) = \sin A \cos B + \cos A \sin B\), the expression equals \(\sin(\theta + 90^\circ - \theta)\).
• This simplifies to \(\sin(90^\circ)\), which is equal to 1.

Both methods confirm the result is 1.

Revision Table: Key Trigonometric Identities

Mastering trigonometric identities is crucial for simplifying expressions and solving trigonometric equations. Here's a quick table of commonly used identities.

Additional Information on Trigonometric Simplification

Simplifying trigonometric expressions is a common task in trigonometry. It often involves transforming a complex expression into a simpler one using known identities and algebraic manipulation. This skill is essential for solving trigonometric equations, evaluating integrals in calculus, and analyzing periodic functions.

When simplifying, look for:

• Opportunities to use reciprocal or quotient identities to express everything in terms of sine and cosine.
• Forms that match Pythagorean identities.
• Angles like \(90^\circ \pm \theta\), \(180^\circ \pm \theta\), \(270^\circ \pm \theta\), \(360^\circ \pm \theta\) where reduction formulas or complementary/supplementary angle identities can be applied.
• Forms that match sum, difference, double, or half angle identities.
• Common factors or algebraic patterns (like factoring differences of squares).

Practice is key to becoming proficient in recognizing which identity to use in different situations.