Problem

Perform the indicated operation and simplify the result. \[ \frac{\cos \theta}{\sec \theta}+\frac{\sin \theta}{\csc \theta} \] What does $\frac{\cos \theta}{\sec \theta}+\frac{\sin \theta}{\csc \theta}$ equal? A. $2 \sin \theta \cos \theta$ B. $\sin \theta+\cos \theta$ C. $\csc \theta$ D. 1 E. $\sec \theta$ F. 0

Solution

Step 1 :Given the expression \(\frac{\cos \theta}{\sec \theta}+\frac{\sin \theta}{\csc \theta}\)

Step 2 :Recall that the secant of an angle in a right triangle is the reciprocal of the cosine function, and the cosecant of an angle is the reciprocal of the sine function.

Step 3 :Therefore, we can rewrite the given expression as \(\cos \theta \cdot \cos \theta + \sin \theta \cdot \sin \theta\)

Step 4 :Which simplifies to \(\sin^2(\theta) + \cos^2(\theta)\)

Step 5 :According to the Pythagorean identity in trigonometry, \(\sin^2(\theta) + \cos^2(\theta) = 1\)

Step 6 :Final Answer: \(\boxed{1}\)

From Solvely APP
Source: https://solvelyapp.com/problems/19293/

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