Answer:
See below for proof.
Step-by-step explanation:
Given trigonometric identity:
\(\dfrac{\sec(x)}{\sin(x)}-\dfrac{\sin(x)}{\cos(x)}=\cot(x)\)
To prove the given trigonometric identity, manipulate the left side until it simplifies into a form equivalent to the right side, using known trigonometric identities and algebraic manipulations.
Begin by rewriting sec(x) in terms of cos(x) using the secant reciprocal identity:
\(\dfrac{\frac{1}{\cos(x)}}{\sin(x)}-\dfrac{\sin(x)}{\cos(x)}\)
Now, express both fractions with a common denominator:
\(\dfrac{\frac{1}{\cos(x)}\cdot \cos(x)}{\sin(x)\cdot \cos(x)}-\dfrac{\sin(x)\cdot \sin(x)}{\cos(x)\cdot \sin(x)}\\\\\\\\\dfrac{1}{\sin(x)\cos(x)}-\dfrac{\sin^2(x)}{\sin(x)\cos(x)}\)
Combine fractions:
\(\dfrac{1-\sin^2(x)}{\sin(x)\cos(x)}\)
Recall the Pythagorean identity sin²(x) + cos²(x) = 1. Therefore, 1 - sin²(x) = cos²(x). Substituting this into the expression gives:
\(\dfrac{\cos^2(x)}{\sin(x)\cos(x)}\)
Cancel the common factor cos(x):
\(\dfrac{\cos(x)}{\sin(x)}\)
Finally, apply the cotangent quotient identity, cot(x) = cos(x) / sin(x):
\(\cot(x)\)
Therefore, the left side is equal to cot(x), proving the identity.
