Question

complete the proof of the identity by choosing the rule that justifies each step.
$(1 - \sin^{2}x)\tan x=\cos x\sin x$
to see a detailed description of a rule, select the more information button to the right of the rule.
$(1 - \sin^{2}x)\tan x$
$=\cos^{2}x\tan x$
$=\cos^{2}x(\frac{\sin x}{\cos x})$
$=\cos x\sin x$

Explanation:

Step1: Use Pythagorean identity

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

Step2: Use Quotient identity

The Quotient identity for tangent is \(\tan x=\frac{\sin x}{\cos x}\).

Step3: Simplify the expression

$$
LATEXBLOCK0
$$

Answer:

The first step uses the Pythagorean rule, the second step uses the Quotient rule and the third step uses Algebra (simplification of the expression \(\cos^{2}x\times\frac{\sin x}{\cos x}\) by canceling one factor of \(\cos x\)).