Question

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

$$\begin{array} { l l } { \\text { statement } } & { \\text { rule } } \\\\ { \\cot x \\left( \\sec ^ { 2 } x - 1 \ ight) } & { } \\\\ { = \\frac { \\cos x } { \\sin x } \\left( \\sec ^ { 2 } x - 1 \ ight) } & { \\text { rule? } } \\\\ { = \\frac { \\cos x } { \\sin x } \\left( \\tan ^ { 2 } x \ ight) } & { \\text { rule? } } \\\\ { = \\frac { \\cos x } { \\sin x } \\left( \\frac { \\sin ^ { 2 } x } { \\cos ^ { 2 } x } \ ight) } & { \\text { rule? } } \\\\ { = \\frac { \\sin x } { \\cos x } } & { \\text { rule? } } \\\\ { = \\tan x } & { \\text { rule? } } \\end{array}$$

\\)

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.