Understanding the Trigonometry Problem

The question asks us to find the value of the expression \(1 - \sin 35^\circ \cos 55^\circ\). This involves trigonometric functions of specific angles. To simplify this expression and find its value, we can use trigonometric identities, specifically complementary angle identities and the Pythagorean identity.

Applying Complementary Angle Identities

Complementary angles are two angles that add up to \(90^\circ\). There are relationships between the trigonometric functions of complementary angles. One key identity is:

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

We can apply this identity to \(\cos 55^\circ\). Notice that \(35^\circ + 55^\circ = 90^\circ\), so \(35^\circ\) and \(55^\circ\) are complementary angles. Therefore, \(55^\circ = 90^\circ - 35^\circ\).

Using the identity:

\(\cos 55^\circ = \sin (90^\circ - 55^\circ) = \sin 35^\circ\)

Simplifying the Expression

Now we can substitute \(\cos 55^\circ = \sin 35^\circ\) back into the original expression \(1 - \sin 35^\circ \cos 55^\circ\):

Expression \( = 1 - \sin 35^\circ \cdot (\sin 35^\circ) \)

Expression \( = 1 - \sin^2 35^\circ \)

Using the Pythagorean Identity

The fundamental Pythagorean trigonometric identity is:

\(\sin^2 \theta + \cos^2 \theta = 1\)

We can rearrange this identity to solve for \(\cos^2 \theta\):

\(\cos^2 \theta = 1 - \sin^2 \theta\)

Applying this identity with \(\theta = 35^\circ\), we get:

\(1 - \sin^2 35^\circ = \cos^2 35^\circ\)

Final Value of the Expression

So, the value of the expression \(1 - \sin 35^\circ \cos 55^\circ\) simplifies to \(\cos^2 35^\circ\).

Comparing with Options

Let's compare our simplified result with the given options:

• Option 1: \(\sec^2 55^\circ\)
• Option 2: \(\csc^2 55^\circ\)
• Option 3: \(\cos^2 35^\circ\)
• Option 4: \(\sin^2 35^\circ\)

Our result, \(\cos^2 35^\circ\), matches Option 3.

The step-by-step calculation is:

Revision Table: Key Trigonometric Identities

Additional Information on Trigonometric Simplification

Simplifying trigonometric expressions often involves recognizing opportunities to apply fundamental identities. The complementary angle identities relate the sine and cosine of acute angles in a right triangle, and similar relationships exist for tangent/cotangent and secant/cosecant. The Pythagorean identity is crucial as it links the squares of sine and cosine, allowing conversion between them. Mastering these identities is key to solving many trigonometry problems.

Remember that complementary angle identities work for any angle \(\theta\):

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

The Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) can also be rearranged or divided by \(\sin^2 \theta\) or \(\cos^2 \theta\) to get other useful forms:

• \(1 + \tan^2 \theta = \sec^2 \theta\)
• \(1 + \cot^2 \theta = \csc^2 \theta\)

These identities are foundational for simplifying complex trigonometric expressions and solving equations.