Question

complete the proof of the identity by choosing the rule that justifies each step.

(sec²x - 1)csc²x = sec²x

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statement

(sec²x - 1)csc²x

= tan²x csc²x

= (sin²x / cos²x)csc²x

= (sin²x / cos²x)(1 / sin²x)

= 1 / cos²x

= sec²x

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.