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Chapter 5: Problem 44

Verify each identity. $$\frac{\cot x+\cot y}{1-\cot x \cot y}=\frac{\cos x \sin y+\sin x \cos y}{\sin x \sin y-\cos x \cos y}$$

Short Answer

Expert verified
The given trigonometric identity has been successfully verified through the process of translating the cotangent terms into equivalent sine and cosine terms and simplifying the resulting equation.

Step by step solution

01

Expanding the identities

We start by converting the \(\cot\) terms to their equivalent \(\cos/\sin\) terms:$$\frac{\cot x+\cot y}{1-\cot x \cot y}=\frac{\frac{\cos x }{\sin x }+\frac{\cos y }{\sin y}}{1-\frac{\cos x }{\sin x }*\frac{\cos y }{\sin y}}.$$
02

Simplify the expression

Next, we find the common denominator to simplify the fractions:$$=\frac{\sin y \cos x + \sin x \cos y}{\sin x \sin y-\cos x \cos y}$$. The numerator and the denominator of this fraction match the ones from the RHS of the original equation.
03

Verification

As both sides of the equation are now the same, we have successfully verified the identity: $$\frac{\cot x+\cot y}{1-\cot x \cot y}=\frac{\cos x \sin y+\sin x \cosy}{\sin x \sin y-\cos x \cos y}$$.

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Most popular questions from this chapter

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$\tan x \sec x=2 \tan x$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning.After using an identity to determine the exact value of \(\sin 105^{\circ}, 1\) verified the result with a calculator.

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Use a graphing utility to approximate the solutions of each equation in the interval \([0,2 \pi) .\) Round to the nearest hundredth of a radian. $$\sin 2 x=2-x^{2}$$
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