complete the proof of the identity by choosing the rule that justifies each step.\n\n(sec²x - 1)csc²x =…
complete the proof of the identity by choosing the rule that justifies each step.\n\n(sec²x - 1)csc²x = sec²x\n\nto see a detailed description of a rule, select the more information button to the right of the rule.\n\nstatement\n\n(sec²x - 1)csc²x\n\n= tan²x csc²x\n\n= (sin²x / cos²x)csc²x\n\n= (sin²x / cos²x)(1 / sin²x)\n\n= 1 / cos²x\n\n= sec²x
Answer
Explanation:
Step1: Use Pythagorean identity
We know that (1+\tan^{2}x = \sec^{2}x), so (\sec^{2}x-1=\tan^{2}x).
Step2: Use quotient identity
The quotient identity is (\tan x=\frac{\sin x}{\cos x}), so (\tan^{2}x=\frac{\sin^{2}x}{\cos^{2}x}).
Step3: Use reciprocal identity
The reciprocal identity is (\csc x=\frac{1}{\sin x}), so (\csc^{2}x=\frac{1}{\sin^{2}x}).
Step4: Simplify the expression
(\frac{\sin^{2}x}{\cos^{2}x}\cdot\frac{1}{\sin^{2}x}=\frac{1}{\cos^{2}x}) (by canceling out (\sin^{2}x)).
Step5: Use reciprocal identity
Since (\sec x=\frac{1}{\cos x}), then (\frac{1}{\cos^{2}x}=\sec^{2}x).
Answer:
The rules (from top - to - bottom) are: Pythagorean identity, Quotient identity, Reciprocal identity, Simplify (by canceling), Reciprocal identity.