trigonometric identities and equations\nverifying a trigonometric identity: problem type 2\ncomplete the…

trigonometric identities and equations\nverifying a trigonometric identity: problem type 2\ncomplete the proof of the identity by choosing the rule that justifies each step.\n$(\\csc ^{2}x - 1)\\sin ^{2}x = \\cos ^{2}x$\nto see a detailed description of a rule, select the more information button to the right of the rule.\nstatement\n$(\\csc ^{2}x - 1)\\sin ^{2}x$\n$=(\\cot ^{2}x)\\sin ^{2}x$\n$=(\\frac{\\cos ^{2}x}{\\sin ^{2}x})\\sin ^{2}x$\n$=\\cos ^{2}x$\nrule\nrule?\nrule?\nrule?

trigonometric identities and equations\nverifying a trigonometric identity: problem type 2\ncomplete the proof of the identity by choosing the rule that justifies each step.\n$(\\csc ^{2}x - 1)\\sin ^{2}x = \\cos ^{2}x$\nto see a detailed description of a rule, select the more information button to the right of the rule.\nstatement\n$(\\csc ^{2}x - 1)\\sin ^{2}x$\n$=(\\cot ^{2}x)\\sin ^{2}x$\n$=(\\frac{\\cos ^{2}x}{\\sin ^{2}x})\\sin ^{2}x$\n$=\\cos ^{2}x$\nrule\nrule?\nrule?\nrule?

Answer

Explanation:

Step 1: Use Pythagorean Identity

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

Step 2: Use Quotient Identity

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

Step 3: Simplify the Expression

We have (\left(\frac{\cos^{2}x}{\sin^{2}x}\right)\sin^{2}x). When we multiply (\frac{\cos^{2}x}{\sin^{2}x}) by (\sin^{2}x), the (\sin^{2}x) terms cancel out ((\frac{\cos^{2}x}{\sin^{2}x}\times\sin^{2}x=\cos^{2}x)).

Answer:

The rules are: Pythagorean Identity, Quotient Identity, Simplify (by canceling (\sin^{2}x) terms).