Question

complete the proof of the identity by choosing the rule that justifies each step.
(cot x(sec ^{2}x - 1)=\tan x)
to see a detailed description of a rule, select the more information button to the right of the rule.

(cot x(sec ^{2}x - 1))
(=\frac{cos x}{sin x}(sec ^{2}x - 1)) rule?
(=\frac{cos x}{sin x}(\tan ^{2}x)) rule?
(=\frac{cos x}{sin x}(\frac{sin ^{2}x}{cos ^{2}x})) rule?
(=\frac{sin x}{cos x}) rule?
(=\tan x) rule?

Explanation:

Step1: Use the definition of cotangent

\(\cot x=\frac{\cos x}{\sin x}\)

Step2: Use the Pythagorean identity \(\sec^{2}x - 1=\tan^{2}x\)

\(\sec^{2}x-1=\tan^{2}x\)

Step3: Use the definition of tangent \(\tan x=\frac{\sin x}{\cos x}\)

\(\tan^{2}x=\frac{\sin^{2}x}{\cos^{2}x}\)

Step4: Simplify the fraction

\(\frac{\cos x}{\sin x}\times\frac{\sin^{2}x}{\cos^{2}x}=\frac{\sin x}{\cos x}\)

Step5: Use the definition of tangent \(\tan x=\frac{\sin x}{\cos x}\)

\(\frac{\sin x}{\cos x}=\tan x\)

Answer:

The proof is completed by using the definitions of trigonometric functions (\(\cot x=\frac{\cos x}{\sin x}\), \(\tan x=\frac{\sin x}{\cos x}\)) and the Pythagorean identity \(\sec^{2}x - 1=\tan^{2}x\) at each step.