Problem 60
Question
Verify each identity. \(\frac{\sin x+\cos x}{\sin x}-\frac{\cos x-\sin x}{\cos x}=\sec x \csc x\)
Step-by-Step Solution
Verified Answer
\(\frac{\sin x+\cos x}{\sin x}-\frac{\cos x-\sin x}{\cos x}\) simplifies to \(\sec x \csc x\), hence the original identity is verified.
1Step 1: Divide the Numerators by Denominators
Begin by simplifying each fraction involving sine and cosine functions on the left side of the equation \(\frac{\sin x+\cos x}{\sin x}-\frac{\cos x-\sin x}{\cos x} =1+ \frac{\cos x}{\sin x} -1 + \frac{\sin x}{\cos x}= \frac{\cos x}{\sin x} + \frac{\sin x}{\cos x}\)
2Step 2: Apply Trigonometric Identities
The second step involves substituting the reciprocals of both the sine and cosine functions, which will lead to the secant and cosecant identities: \(\frac{1}{\sin x}+\frac{1}{\cos x}\)
3Step 3: Match Both Sides
After the manipulations, one can observe that both sides of the equation now match. Therefore, it can be deduced that the original identity provided is indeed correct.
Key Concepts
Verifying Trigonometric IdentitiesSine and Cosine FunctionsSecant and Cosecant Identities
Verifying Trigonometric Identities
The process of verifying trigonometric identities often involves a sequence of algebraic and trigonometric steps to show that two expressions are equivalent. This can include factoring, finding common denominators, and using known trigonometric identities such as reciprocal identities and Pythagorean identities.
When seeking to prove an identity like \( \frac{\sin x+\cos x}{\sin x}-\frac{\cos x-\sin x}{\cos x}=\sec x \csc x \), the first step is to simplify each term separately. For the left-side expression, dividing the numerators by the respective denominators simplifies to individual terms — some of which cancel each other out because they are additive inverses. What remains are terms that involve the reciprocal of sine and cosine functions. This leads to an expression involving secant and cosecant, which are directly related to sine and cosine through their identities. The aim is to manipulate the terms until they look exactly like the right side of the original identity. When both sides match, the identity is verified.
To enhance understanding, visual aids illustrating the unit circle and the definitions of sine, cosine, secant, and cosecant can be helpful.
When seeking to prove an identity like \( \frac{\sin x+\cos x}{\sin x}-\frac{\cos x-\sin x}{\cos x}=\sec x \csc x \), the first step is to simplify each term separately. For the left-side expression, dividing the numerators by the respective denominators simplifies to individual terms — some of which cancel each other out because they are additive inverses. What remains are terms that involve the reciprocal of sine and cosine functions. This leads to an expression involving secant and cosecant, which are directly related to sine and cosine through their identities. The aim is to manipulate the terms until they look exactly like the right side of the original identity. When both sides match, the identity is verified.
To enhance understanding, visual aids illustrating the unit circle and the definitions of sine, cosine, secant, and cosecant can be helpful.
Sine and Cosine Functions
Sine (\(\sin\)) and cosine (\(\cos\)) functions are fundamental in trigonometry, defined for any angle on the unit circle. They represent the y-coordinate and x-coordinate of a point on the unit circle, respectively.
In practice, algebraic manipulation becomes necessary to simplify and compare trigonometric expressions. When dealing with fractions, such as in the exercise \(\frac{\sin x+\cos x}{\sin x}-\frac{\cos x-\sin x}{\cos x}\), recognizing when a term can be split into simpler fractions, or combined through common denominators, is a strategy that simplifies the verification process.
- \(\sin x\) corresponds to the vertical displacement from the x-axis.
- \(\cos x\) corresponds to the horizontal displacement from the y-axis.
In practice, algebraic manipulation becomes necessary to simplify and compare trigonometric expressions. When dealing with fractions, such as in the exercise \(\frac{\sin x+\cos x}{\sin x}-\frac{\cos x-\sin x}{\cos x}\), recognizing when a term can be split into simpler fractions, or combined through common denominators, is a strategy that simplifies the verification process.
Secant and Cosecant Identities
Secant (\(\sec\)) and cosecant (\(\csc\)) are less common but equally important trigonometric functions, defined as the reciprocal of cosine and sine, respectively. Here are their identities:
When you encounter secant or cosecant in an exercise, you should immediately think of their relationship to cosine and sine as this will often lead to simplification opportunities. For example, multiplying by the reciprocal function can help combine terms or facilitate other algebraic manipulations, ultimately helping to verify identities and solve equations involving these functions.
- \(\sec x = \frac{1}{\cos x}\)
- \(\csc x = \frac{1}{\sin x}\)
When you encounter secant or cosecant in an exercise, you should immediately think of their relationship to cosine and sine as this will often lead to simplification opportunities. For example, multiplying by the reciprocal function can help combine terms or facilitate other algebraic manipulations, ultimately helping to verify identities and solve equations involving these functions.
Other exercises in this chapter
Problem 60
verify each identity. $$ \frac{\sin 2 x+(\sin 3 x+\sin x)}{\cos 2 x+(\cos 3 x+\cos x)}=\tan 2 x $$
View solution Problem 60
Verify each identity. $$ \sin ^{2} \frac{\theta}{2}=\frac{\csc \theta-\cot \theta}{2 \csc \theta} $$
View solution Problem 61
Find the exact value of the following under the given conditions: a. \(\cos (\alpha+\beta)\) b. \(\sin (\alpha+\beta)\) c. \(\tan (\alpha+\beta)\) \(\cos \alpha
View solution Problem 61
Solve each equation on the interval \([0,2 \pi)\) $$ \tan ^{2} x \cos x=\tan ^{2} x $$
View solution