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Simplify the following expression:
[tex]\[ \frac{\sin \theta \div \sec \theta}{\cos \theta \div \sin \theta} \][/tex]


Sagot :

Alright, let's break down the given mathematical expression step-by-step.

Given expression:
[tex]\[ \frac{\left(\frac{\sin \theta}{\sec \theta}\right)}{\left(\frac{\cos \theta}{\sin \theta}\right)} \][/tex]

### Step 1: Simplify the inner terms
First, let's simplify each component inside the parentheses.

1. Simplify [tex]\(\frac{\sin \theta}{\sec \theta}\)[/tex]:
- Recall that [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex].
- So, [tex]\(\frac{\sin \theta}{\sec \theta} = \sin \theta \times \cos \theta\)[/tex].

Hence,
[tex]\[ \frac{\sin \theta}{\sec \theta} = \sin \theta \cos \theta \][/tex]

2. Simplify [tex]\(\frac{\cos \theta}{\sin \theta}\)[/tex]:
- This is a straightforward simplification.

Hence,
[tex]\[ \frac{\cos \theta}{\sin \theta} = \frac{\cos \theta}{\sin \theta} \][/tex]

### Step 2: Combine the simplified terms
Next, we rewrite the original expression with these simplified terms.

So the expression becomes:
[tex]\[ \frac{\sin \theta \cos \theta}{\frac{\cos \theta}{\sin \theta}} \][/tex]

### Step 3: Simplify the combined fraction
To simplify this, we multiply by the reciprocal of the denominator.

[tex]\[ \frac{\sin \theta \cos \theta}{\frac{\cos \theta}{\sin \theta}} = \sin \theta \cos \theta \times \frac{\sin \theta}{\cos \theta} \][/tex]

### Step 4: Simplify the multiplication
Now, we can cancel out the common terms [tex]\(\cos \theta\)[/tex]:

[tex]\[ \left(\sin \theta \cos \theta\right) \times \left(\frac{\sin \theta}{\cos \theta}\right) = \sin^2 \theta \][/tex]

Finally, the simplified form of the given expression is:
[tex]\[ \sin^2 \theta \][/tex]

Hence, our final answer is: [tex]\(\sin^2 \theta\)[/tex].
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