Math Problem Statement

Simplify the trigonometric expressions: 15. sin(x) sec(x) / tan(x), 16. cos(x) sec(x) / cot(x), 17. (sin(t) + tan(t)) / tan(t), 18. (1 + cot(A)) / csc(A).

Solution

Let's simplify the given trigonometric expressions step by step:

15. sinxsecxtanx\frac{\sin x \sec x}{\tan x}

  1. Recall that secx=1cosx\sec x = \frac{1}{\cos x} and tanx=sinxcosx\tan x = \frac{\sin x}{\cos x}.
  2. Substituting these definitions: sinxsecxtanx=sinx1cosxsinxcosx.\frac{\sin x \sec x}{\tan x} = \frac{\sin x \cdot \frac{1}{\cos x}}{\frac{\sin x}{\cos x}}.
  3. Simplify the numerator and denominator: sinx1cosxsinxcosx=sinxcosxcosxsinx.\frac{\sin x \cdot \frac{1}{\cos x}}{\frac{\sin x}{\cos x}} = \frac{\sin x}{\cos x} \cdot \frac{\cos x}{\sin x}.
  4. Cancel out sinx\sin x and cosx\cos x where appropriate: sinxcosxcosxsinx=1.\frac{\sin x}{\cos x} \cdot \frac{\cos x}{\sin x} = 1. Answer: 1

16. cosxsecxcotx\frac{\cos x \sec x}{\cot x}

  1. Recall that secx=1cosx\sec x = \frac{1}{\cos x} and cotx=cosxsinx\cot x = \frac{\cos x}{\sin x}.
  2. Substitute these definitions: [ \frac{\cos x \sec x}{\cot x} =

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Simplification of Trigonometric Expressions

Formulas

sec(x) = 1 / cos(x)
tan(x) = sin(x) / cos(x)
cot(x) = cos(x) / sin(x)
csc(x) = 1 / sin(x)

Theorems

Trigonometric Identities

Suitable Grade Level

Grades 10-12