Solving the Trigonometric Identity Problem

We are given the equation \(\sec \theta + \cos \theta = 32\) and asked to find the value of \(\sec^2 \theta + \cos^2 \theta\).

To find \(\sec^2 \theta + \cos^2 \theta\), we can consider squaring the given equation. Remember the algebraic identity: \((a+b)^2 = a^2 + 2ab + b^2\).

Let's square both sides of the equation \(\sec \theta + \cos \theta = 32\):

\((\sec \theta + \cos \theta)^2 = (32)^2\)

Applying the identity on the left side:

\(\sec^2 \theta + 2(\sec \theta)(\cos \theta) + \cos^2 \theta = 32^2\)

Now, let's simplify the term \((\sec \theta)(\cos \theta)\). We know that the secant function is the reciprocal of the cosine function. That is, \(\sec \theta = \frac{1}{\cos \theta}\).

So, \((\sec \theta)(\cos \theta) = \left(\frac{1}{\cos \theta}\right)(\cos \theta)\).

Assuming \(\cos \theta \neq 0\), the term simplifies to:

\((\sec \theta)(\cos \theta) = 1\)

Substitute this value back into the squared equation:

\(\sec^2 \theta + 2(1) + \cos^2 \theta = 32^2\)

\(\sec^2 \theta + 2 + \cos^2 \theta = 1024\)

Now, rearrange the equation to isolate the term we want to find, \(\sec^2 \theta + \cos^2 \theta\):

\(\sec^2 \theta + \cos^2 \theta = 1024 - 2\)

\(\sec^2 \theta + \cos^2 \theta = 1022\)

Thus, the value of \(\sec^2 \theta + \cos^2 \theta\) is 1022.

Step-by-Step Calculation

1. Start with the given equation: \(\sec \theta + \cos \theta = 32\).
2. Square both sides of the equation: \((\sec \theta + \cos \theta)^2 = (32)^2\).
3. Expand the left side using \((a+b)^2 = a^2 + 2ab + b^2\): \(\sec^2 \theta + 2(\sec \theta)(\cos \theta) + \cos^2 \theta = 1024\).
4. Use the reciprocal identity \(\sec \theta = \frac{1}{\cos \theta}\) to simplify \((\sec \theta)(\cos \theta)\) to 1.
5. Substitute 1 into the equation: \(\sec^2 \theta + 2(1) + \cos^2 \theta = 1024\).
6. Simplify: \(\sec^2 \theta + 2 + \cos^2 \theta = 1024\).
7. Isolate \(\sec^2 \theta + \cos^2 \theta\) by subtracting 2 from both sides: \(\sec^2 \theta + \cos^2 \theta = 1024 - 2\).
8. Calculate the final value: \(\sec^2 \theta + \cos^2 \theta = 1022\).

Comparing with Options

The calculated value 1022 matches Option 1.

Revision Table: Key Trigonometric Concepts

Additional Information: Understanding Reciprocal Identities

Reciprocal identities in trigonometry relate the six trigonometric functions to each other. They are fundamental for simplifying expressions and solving equations.

• \(\sin \theta\) and \(\csc \theta\) are reciprocals: \(\csc \theta = \frac{1}{\sin \theta}\) (when \(\sin \theta \neq 0\))
• \(\cos \theta\) and \(\sec \theta\) are reciprocals: \(\sec \theta = \frac{1}{\cos \theta}\) (when \(\cos \theta \neq 0\))
• \(\tan \theta\) and \(\cot \theta\) are reciprocals: \(\cot \theta = \frac{1}{\tan \theta}\) (when \(\tan \theta \neq 0\))

In this problem, the identity \(\sec \theta \cdot \cos \theta = 1\) (derived from \(\sec \theta = 1/\cos \theta\)) was crucial for simplifying the middle term after squaring the expression.

These identities are often used in combination with Pythagorean identities (\(\sin^2 \theta + \cos^2 \theta = 1\), etc.) and other trigonometric formulas to solve a wide range of problems.