simplify. 12. (sec^(2)x-1)/(sec x-1)

Question

simplify. 12.  (sec^(2)x-1)/(sec x-1)

Bytelearn · Accepted Answer

Recognize Pythagorean Identity: We recognize that the numerator sec^(2)x - 1 is a Pythagorean identity which can be rewritten as tan^(2)x. sec^(2)x - 1 = tan^(2)xRewrite Using Identity: Now we rewrite the original expression using the identity from step 1. (12. (sec^(2)x-1))/(sec x-1) = (12. tan^(2)x)/(sec x-1)Factor Numerator: We can factor the numerator as 12 * tan^(2)x, which is 12 times the square of tan(x). (12. tan^(2)x)/(sec x-1) = 12 * (tan^(2)x)/(sec x-1)Substitute Trigonometric Ratios: We notice that tan(x) can be expressed as sin(x)/cos(x) and sec(x) as 1/cos(x). So, tan^(2)x = (sin(x)/cos(x))^2 and sec x = 1/cos(x)Simplify Expression: Substitute tan(x) and sec(x) with their respective trigonometric ratios in the expression. 12 * (tan^(2)x)/(sec x-1) = 12 * ((sin(x)/cos(x))^2)/((1/cos(x))-1)Rewrite Using Pythagorean Identity: Simplify the expression by multiplying both the numerator and the denominator by cos(x), the common denominator of the fractions within the expression. 12 * ((sin(x)/cos(x))^2)/((1/cos(x))-1) = 12 * (sin^(2)x)/(1 - cos(x))Factor Numerator: We recognize that sin^(2)x can be rewritten using the Pythagorean identity sin^(2)x = 1 - cos^(2)x. 12 * (sin^(2)x)/(1 - cos(x)) = 12 * (1 - cos^(2)x)/(1 - cos(x))Cancel Common Factor: Now we can factor the numerator as a difference of squares: (1 - cos(x))(1 + cos(x)). 12 * (1 - cos^(2)x)/(1 - cos(x)) = 12 * ((1 - cos(x))(1 + cos(x)))/(1 - cos(x))Final Simplified Form: We can cancel out the common factor of (1 - cos(x)) in the numerator and the denominator. 12 * ((1 - cos(x))(1 + cos(x)))/(1 - cos(x)) = 12 * (1 + cos(x))The final simplified form of the expression is 12 * (1 + cos(x)).

simplify. $\newline$ $\frac{\sec^{2}x-1}{\sec x-1}$

Full solution

More problems from Simplify radical expressions involving fractions

Related topics

simplify. \newlinesec⁡2x−1sec⁡x−1\frac{\sec^{2}x-1}{\sec x-1}secx−1sec2x−1​

simplify. $\newline$ $\frac{\sec^{2}x-1}{\sec x-1}$