Simplify: (sec^(2)theta)/(sec^(2)theta-1)=csc^(2)theta

Recognize identity: Recognize the trigonometric identity: (sec^(2)theta) = 1 + tan^(2)theta Substitute into equation: Substitute the identity into the equation: (sec^(2)theta)/(sec^(2)theta - 1) = (1 + tan^(2)theta)/(1 + tan^(2)theta - 1) Simplify expression: Simplify the expression: (1 + tan^(2)theta)/(tan^(2)theta) = 1/(sin^(2)theta/cos^(2)theta) Further simplify using cotangent: Further simplify using the identity for cotangent: cot^(2)theta = cos^(2)theta/sin^(2)theta Use Pythagorean identity: Use the Pythagorean identity for cosecant: csc^(2)theta = 1/sin^(2)theta Substitute back to find relationship: Substitute back to find the relationship: 1 + cot^(2)theta = csc^(2)theta

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Simplify: (sec^(2)theta)/(sec^(2)theta-1)=csc^(2)theta

Simplify: $\frac{\sec^{2}\theta}{\sec^{2}\theta-1}=\csc^{2}\theta$

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Algebra 2

Sin, cos, and tan of special angles

Full solution

Q. Simplify:

\frac{\sec^{2}\theta}{\sec^{2}\theta-1}=\csc^{2}\theta

Recognize identity: Recognize the trigonometric identity: $\sec^{2}\theta = 1 + \tan^{2}\theta$
Substitute into equation: Substitute the identity into the equation: $\frac{\sec^{2}\theta}{\sec^{2}\theta - 1} = \frac{1 + \tan^{2}\theta}{1 + \tan^{2}\theta - 1}$
Simplify expression: Simplify the expression: $\newline$ $(1 + \tan^{2}\theta)/(\tan^{2}\theta) = 1/(\sin^{2}\theta/\cos^{2}\theta)$
Further simplify using cotangent: Further simplify using the identity for cotangent: $\cot^{2}\theta = \frac{\cos^{2}\theta}{\sin^{2}\theta}$
Use Pythagorean identity: Use the Pythagorean identity for cosecant: $\csc^2\theta = \frac{1}{\sin^2\theta}$
Substitute back to find relationship: Substitute back to find the relationship: $\newline$ $1 + \cot^{2}\theta = \csc^{2}\theta$