Absolutely, let’s simplify the given expression step-by-step to see how it equals 1.
Given expression:
[tex]\[ \frac{\sin \theta}{\csc \theta} + \frac{\cos \theta}{\sec \theta} \][/tex]
First, recall the definitions of cosecant ([tex]\(\csc\)[/tex]) and secant ([tex]\(\sec\)[/tex]) in terms of sine ([tex]\(\sin\)[/tex]) and cosine ([tex]\(\cos\)[/tex]):
[tex]\[ \csc \theta = \frac{1}{\sin \theta} \][/tex]
[tex]\[ \sec \theta = \frac{1}{\cos \theta} \][/tex]
Substituting these into the given expression:
[tex]\[ \frac{\sin \theta}{\frac{1}{\sin \theta}} + \frac{\cos \theta}{\frac{1}{\cos \theta}} \][/tex]
Now simplify each fraction:
[tex]\[ \frac{\sin \theta}{\frac{1}{\sin \theta}} = \sin \theta \cdot \sin \theta = \sin^2 \theta \][/tex]
[tex]\[ \frac{\cos \theta}{\frac{1}{\cos \theta}} = \cos \theta \cdot \cos \theta = \cos^2 \theta \][/tex]
Now substitute these back into the expression:
[tex]\[ \sin^2 \theta + \cos^2 \theta \][/tex]
Using the Pythagorean identity, which states:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
Thus, the simplified expression is:
[tex]\[ 1 \][/tex]
Therefore,
[tex]\[ \frac{\sin \theta}{\csc \theta} + \frac{\cos \theta}{\sec \theta} = 1 \][/tex]
And we’ve shown that the original expression simplifies to 1.
