Simplify. csc(x) sec(x) Use algebra and the fundamental trigonometric identities. Your answer should be a number or use a single trigonometric function. cot(y) sin(x) cos(D) tan(D) cot(D) sec(D)
The Correct Answer and Explanation is:
Correct Answer: cot(x)
Explanation:
The problem asks to simplify the trigonometric expression csc(x) / sec(x) into a single trigonometric function. This can be achieved by using fundamental trigonometric identities, specifically the reciprocal and quotient identities.
Step 1: Apply Reciprocal Identities
The first step is to express csc(x) and sec(x) in terms of the more basic functions, sin(x) and cos(x). We use the reciprocal identities:
- The cosecant function is the reciprocal of the sine function: csc(x) = 1 / sin(x)
- The secant function is the reciprocal of the cosine function: sec(x) = 1 / cos(x)
By substituting these identities into the original expression, we get:
csc(x) / sec(x) = (1 / sin(x)) / (1 / cos(x))
Step 2: Simplify the Complex Fraction
The expression is now a complex fraction (a fraction where the numerator, denominator, or both contain fractions). To simplify this, we use the algebraic rule for dividing fractions: dividing by a fraction is equivalent to multiplying by its reciprocal.
The reciprocal of the denominator (1 / cos(x)) is (cos(x) / 1). So, we can rewrite the expression as a multiplication:
(1 / sin(x)) * (cos(x) / 1)
Step 3: Perform the Multiplication
Now, we multiply the numerators together and the denominators together:
(1 * cos(x)) / (sin(x) * 1) = cos(x) / sin(x)
Step 4: Apply the Quotient Identity
The final step is to recognize that the resulting expression, cos(x) / sin(x), is the definition of another fundamental trigonometric function. The quotient identity for the cotangent function states:
- cot(x) = cos(x) / sin(x)
Therefore, we can replace cos(x) / sin(x) with cot(x).
Conclusion
By systematically applying the reciprocal and quotient identities, we have simplified the expression csc(x) / sec(x) to a single trigonometric function, cot(x). The answer satisfies the condition of being a single function.
