Sure, let's analyze the equation [tex]\( 1 = \sec(\beta) \cdot \cos(\beta) \)[/tex].
Step 1: Understand the Trigonometric Identities
First, remember the definition of the secant function. By definition:
[tex]\[ \sec(\beta) = \frac{1}{\cos(\beta)} \][/tex]
Step 2: Substitute the Definition into the Equation
Now, substitute the definition of [tex]\(\sec(\beta)\)[/tex] into the given equation:
[tex]\[ 1 = \sec(\beta) \cdot \cos(\beta) \][/tex]
[tex]\[ 1 = \left( \frac{1}{\cos(\beta)} \right) \cdot \cos(\beta) \][/tex]
Step 3: Simplify the Expression
Next, simplify the right-hand side of the equation. Notice that [tex]\(\frac{1}{\cos(\beta)} \cdot \cos(\beta)\)[/tex] is essentially multiplying a number by its reciprocal:
[tex]\[ 1 = \left( \frac{1}{\cos(\beta)} \right) \cdot \cos(\beta) \][/tex]
[tex]\[ 1 = 1 \][/tex]
Step 4: Conclude the Validity
The equation simplifies to:
[tex]\[ 1 = 1 \][/tex]
Since this statement is always true, it indicates that our original equation:
[tex]\[ 1 = \sec(\beta) \cdot \cos(\beta) \][/tex]
is always true for any value of [tex]\(\beta\)[/tex].
Therefore, the equation [tex]\( 1 = \sec(\beta) \cdot \cos(\beta) \)[/tex] holds true for any value of [tex]\(\beta\)[/tex] where [tex]\(\cos(\beta)\)[/tex] is defined (i.e., [tex]\(\cos(\beta) \neq 0\)[/tex]).
