complete the proof of the identity by choosing the rule that justifies each step. \n\n$$ ( csc ^ { 2 } x - 1…

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

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

Answer

Explanation:

Step1: Use Pythagorean identity

We know that (1 + \cot^{2}x=\csc^{2}x), so (\csc^{2}x - 1=\cot^{2}x).

Step2: Use quotient identity

The quotient identity is (\cot x=\frac{\cos x}{\sin x}), so (\cot^{2}x=\frac{\cos^{2}x}{\sin^{2}x}).

Step3: Simplify the expression

(\left(\frac{\cos^{2}x}{\sin^{2}x}\right)\sin^{2}x=\cos^{2}x) (by canceling out (\sin^{2}x) in the numerator and denominator).

Answer:

The rules are: Pythagorean identity, Quotient identity, Simplify (by canceling).