Question

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.(sec²θ - 1)/(csc²θ - 1)

Accepted Answer

Recall the definitions of the secant and cosecant functions in terms of sine and cosine: $$\sec 	heta = \frac{1}{\cos 	heta}$$ and $$\csc 	heta = \frac{1}{\sin 	heta}. $$Rewrite the given expression $$\frac{\sec$$^{2}$$ 	heta - 1}{\csc$$^{2}$$ 	heta - 1} by $$substituting $$\sec$$^{2}$$ 	heta$$ with $$\frac{1}{\cos$$^{2}$$ 	heta}$$ and $$\csc$$^{2}$$ 	heta$$ with $$\frac{1}{\sin$$^{2}$$ 	heta}$$, so it becomes $$\frac{\frac{1}{\cos$$^{2}$$ 	heta} - 1}{\frac{1}{\sin$$^{2}$$ 	heta} - 1}. $$Simplify the numerator and denominator separately by combining terms over a common denominator: numerator becomes $$\frac{1 - \cos$$^{2}$$ 	heta}{\cos$$^{2}$$ 	heta}$$ and denominator becomes $$\frac{1 - \sin$$^{2}$$ 	heta}{\sin$$^{2}$$ 	heta}. $$Use the Pythagorean identity $$\sin$$^{2}$$ 	heta + \cos$$^{2}$$ 	heta = 1 to $$rewrite $$1 - \cos$$^{2}$$ 	heta as$$ $$\sin$$^{2}$$ 	heta$$ and $$1 - \sin$$^{2}$$ 	heta as$$ $$\cos$$^{2}$$ 	heta. $$Substitute these back into the expression to get $$\frac{\frac{\sin$$^{2}$$ 	heta}{\cos$$^{2}$$ 	heta}}{\frac{\cos$$^{2}$$ 	heta}{\sin$$^{2}$$ 	heta}}$$, then simplify by multiplying numerator by the reciprocal of the denominator to eliminate quotients and express the entire expression in terms of sine and cosine only.

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
(sec²θ - 1)/(csc²θ - 1)

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Key Concepts

Pythagorean Identities

Reciprocal Trigonometric Functions

Algebraic Simplification of Trigonometric Expressions