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

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

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

Answer

Explanation:

Step1: Use the quotient identity

The quotient identity states that (\cot x=\frac{\cos x}{\sin x}). So, (\cot x(\sec^{2}x - 1)=\frac{\cos x}{\sin x}(\sec^{2}x - 1))

Step2: Use the Pythagorean identity

The Pythagorean identity (\sec^{2}x-1 = \tan^{2}x). Then (\frac{\cos x}{\sin x}(\sec^{2}x - 1)=\frac{\cos x}{\sin x}(\tan^{2}x))

Step3: Use the quotient identity again

Since (\tan x=\frac{\sin x}{\cos x}), then (\tan^{2}x=\frac{\sin^{2}x}{\cos^{2}x}). So (\frac{\cos x}{\sin x}(\tan^{2}x)=\frac{\cos x}{\sin x}\cdot\frac{\sin^{2}x}{\cos^{2}x})

Step4: Simplify the expression

(\frac{\cos x}{\sin x}\cdot\frac{\sin^{2}x}{\cos^{2}x}=\frac{\sin x}{\cos x}) (by canceling out one (\cos x) and one (\sin x) )

Step5: Use the quotient identity

Since (\tan x=\frac{\sin x}{\cos x}), then (\frac{\sin x}{\cos x}=\tan x)

Answer:

The rules for each step (from top - to - bottom) are: Quotient Identity, Pythagorean Identity, Quotient Identity, Simplify (Algebraic simplification), Quotient Identity.