Simplify the following. Show each step: (sin(x)/cos(x))² - 1/cos²(x)

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to simplify: (sin(x)/cos(x))^2 - 1/(cos(x))^2

(sin(x)/cos(x))^2 - 1/(cos(x))^2

=> ((sin x)^2 - 1)/(cos x)^2

(sin x)^2 + (cos x)^2 = 1 => (sin x)^2 - 1 = -(cos x)^2

=> -(cos x)^2/(cos x)^2

=> -1

The simplified form of (sin(x)/cos(x))^2 - 1/(cos(x))^2 = -1

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

The first step when subtracting two fractions is to check if they have common denominator.

In this case, they both have the denominator (cos x)^2, therefore, we can re-write the difference in this way:

[(sin x)^2 - 1]/(cos x)^2

We can use the Pythagorean identity, to replace 1:

(sin x)^2 + (cos x)^2 = 1

We'll re-write the fraction:

[(sin x)^2 - (sin x)^2 - (cos x)^2]/(cos x)^2

We'll eliminate like terms within the brackets:

[- (cos x)^2]/(cos x)^2 = -1

The result of subtracting the given fractions is: (sin x)^2/(cos x)^2 - 1/(cos x)^2 = -1.

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