Let's address the given trigonometric identity step by step and demonstrate its validity.
The identity we need to prove is:
[tex]\[
\frac{\sin \theta + \cos \theta}{\sec \theta + \csc \theta} = \sin \theta \cdot \cos \theta
\][/tex]
### Step 1: Rewrite the given identity
We start by simplifying each term. Recall the definitions of [tex]\(\sec \theta\)[/tex] and [tex]\(\csc \theta\)[/tex]:
[tex]\[
\sec \theta = \frac{1}{\cos \theta} \quad \text{and} \quad \csc \theta = \frac{1}{\sin \theta}
\][/tex]
Therefore, the identity can be rewritten as:
[tex]\[
\frac{\sin \theta + \cos \theta}{\frac{1}{\cos \theta} + \frac{1}{\sin \theta}}
\][/tex]
### Step 2: Simplify the denominator
Combine the terms in the denominator to get a common denominator:
[tex]\[
\sec \theta + \csc \theta = \frac{1}{\cos \theta} + \frac{1}{\sin \theta} = \frac{\sin \theta + \cos \theta}{\sin \theta \cos \theta}
\][/tex]
### Step 3: Simplify the overall expression
Insert the simplified denominator back into the fraction:
[tex]\[
\frac{\sin \theta + \cos \theta}{\frac{\sin \theta + \cos \theta}{\sin \theta \cos \theta}}
\][/tex]
When you divide by a fraction, you multiply by its reciprocal:
[tex]\[
= (\sin \theta + \cos \theta) \times \frac{\sin \theta \cos \theta}{\sin \theta + \cos \theta}
\][/tex]
### Step 4: Cancel out common terms
The [tex]\(\sin \theta + \cos \theta\)[/tex] terms cancel out:
[tex]\[
= \sin \theta \cos \theta
\][/tex]
So, we've shown:
[tex]\[
\frac{\sin \theta + \cos \theta}{\sec \theta + \csc \theta} = \sin \theta \cos \theta
\][/tex]
### Conclusion
The given trigonometric identity is proven to be valid, and both sides are equal:
[tex]\[
\frac{\sin \theta+\cos \theta}{\sec \theta+\operatorname{cosec} \theta} = \sin \theta \cdot \cos \theta
\][/tex]
Therefore, the identity holds true.
