complete the proof of the identity by choosing the rule that justifies each step.\n(1 - cos²x)cot²x =…

complete the proof of the identity by choosing the rule that justifies each step.\n(1 - cos²x)cot²x = cos²x\nto see a detailed description of a rule, select the more information button to the right of the rule.\nstatement rule\n(1 - cos²x)cot²x\n= sin²x cot²x rule?\n= sin²x (cos²x / sin²x) rule?\n= cos²x rule?

complete the proof of the identity by choosing the rule that justifies each step.\n(1 - cos²x)cot²x = cos²x\nto see a detailed description of a rule, select the more information button to the right of the rule.\nstatement rule\n(1 - cos²x)cot²x\n= sin²x cot²x rule?\n= sin²x (cos²x / sin²x) rule?\n= cos²x rule?

Answer

Brief Explanations:

  • For the step ((1-\cos^{2}x)\cot^{2}x=\sin^{2}x\cot^{2}x), we use the Pythagorean identity (\sin^{2}x + \cos^{2}x=1), which can be rewritten as (1-\cos^{2}x=\sin^{2}x).
  • For the step (\sin^{2}x\cot^{2}x=\sin^{2}x\left(\frac{\cos^{2}x}{\sin^{2}x}\right)), we use the quotient identity (\cot x=\frac{\cos x}{\sin x}), so (\cot^{2}x = \frac{\cos^{2}x}{\sin^{2}x}).
  • For the step (\sin^{2}x\left(\frac{\cos^{2}x}{\sin^{2}x}\right)=\cos^{2}x), we use the rule of simplifying fractions (canceling out the (\sin^{2}x) terms in the numerator and denominator).

Answer:

  1. Pythagorean Identity
  2. Quotient Identity
  3. Simplifying Fractions (Cancelation)