complete the proof of the identity by choosing the rule that justifies each step. \n(cot x(sec ^{2}x…

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

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

Answer

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.