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

complete the proof of the identity by choosing the rule that justifies each step. \n\\( \\sin ^ { 2 } x \\left( 1 + \\cot ^ { 2 } x \\right) = 1 \\)\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 } } \\\\ { \\sin ^ { 2 } x \\left( 1 + \\cot ^ { 2 } x \\right) } & { } \\\\ { = \\sin ^ { 2 } x \\left( \\csc ^ { 2 } x \\right) } & { \\text { rule? } } \\\\ { = \\sin ^ { 2 } x \\left( \\frac { 1 } { \\sin ^ { 2 } x } \\right) } & { \\text { rule? } } \\\\ { = 1 } & { \\text { rule? } } \\end{array} \\)

complete the proof of the identity by choosing the rule that justifies each step. \n\\( \\sin ^ { 2 } x \\left( 1 + \\cot ^ { 2 } x \\right) = 1 \\)\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 } } \\\\ { \\sin ^ { 2 } x \\left( 1 + \\cot ^ { 2 } x \\right) } & { } \\\\ { = \\sin ^ { 2 } x \\left( \\csc ^ { 2 } x \\right) } & { \\text { rule? } } \\\\ { = \\sin ^ { 2 } x \\left( \\frac { 1 } { \\sin ^ { 2 } x } \\right) } & { \\text { rule? } } \\\\ { = 1 } & { \\text { rule? } } \\end{array} \\)

Answer

Explanation:

Step1: Use Pythagorean Identity

We know that (1+\cot^{2}x = \csc^{2}x) (Pythagorean Identity: (1+\cot^{2}\theta=\csc^{2}\theta)). So, (\sin^{2}x(1 + \cot^{2}x)=\sin^{2}x(\csc^{2}x))

Step2: Use Reciprocal Identity

Since (\csc x=\frac{1}{\sin x}), then (\csc^{2}x=\frac{1}{\sin^{2}x}). So, (\sin^{2}x(\csc^{2}x)=\sin^{2}x\left(\frac{1}{\sin^{2}x}\right))

Step3: Simplify

When we multiply (\sin^{2}x) and (\frac{1}{\sin^{2}x}), for (\sin x\neq0), (\sin^{2}x\times\frac{1}{\sin^{2}x}=1) (Simplification of fraction: (a\times\frac{1}{a} = 1,a\neq0))

Answer:

The rules are: Pythagorean Identity, Reciprocal Identity, Simplification.