Problem 32
Question
$$ (\sin A+\operatorname{cosec} A)^{2}+(\cos A+\sec A)^{2}=\tan ^{2} A+\cot ^{2} A+7 $$
Step-by-Step Solution
Verified Answer
We started by expressing all functions in terms of sine and cosine, and then simplified the equation using trigonometric identities and algebraic operations. By combining fractions on each side and simplifying further, we have proved the given identity:
\(\frac{2\sin^2 A\cos^2 A + 1}{\sin^2 A \cos^2 A} = \frac{\sin^4 A}{\cos^2 A} + \frac{\cos^4 A}{\sin^2 A} + 7\)
1Step 1: Write all trigonometric functions in terms of sine and cosine
Recall that:
\(\csc A = \frac{1}{\sin A}\),
\(\sec A = \frac{1}{\cos A}\),
\(\tan A = \frac{\sin A}{\cos A}\), and
\(\cot A = \frac{\cos A}{\sin A}\)
Now, substitute these expressions in the given equation to express all functions in terms of sine and cosine:
\((\sin A + \frac{1}{\sin A})^2 + (\cos A + \frac{1}{\cos A})^2 = (\frac{\sin^2 A}{\cos^2 A}) + (\frac{\cos^2 A}{\sin^2 A}) + 7\)
2Step 2: Simplify both sides of the equation
First, simplify the squares on both sides of the equation:
\(\sin^2 A + 2 + \frac{1}{\sin^2 A} + \cos^2 A + 2 + \frac{1}{\cos^2 A} = \frac{\sin^4 A}{\cos^2 A} + \frac{\cos^4 A}{\sin^2 A} + 7\)
Since \(\sin^2 A + \cos^2 A = 1\), we can simplify further:
\(2 + \frac{1}{\sin^2 A} + 2 + \frac{1}{\cos^2 A} = \frac{\sin^4 A}{\cos^2 A} + \frac{\cos^4 A}{\sin^2 A} + 7\)
3Step 3: Combine fractions on each side
Combine the fractions on each side of the equation by finding the common denominators:
\(\frac{\sin^2 A + 2\sin^4 A + \cos^2 A + 2\cos^4 A}{\sin^2 A \cos^2 A} = \frac{\sin^4 A}{\cos^2 A} + \frac{\cos^4 A}{\sin^2 A} + 7\)
Now, simplify the left side of the equation by factoring out \(2\sin^2 A \cos^2 A\):
\(\frac{2\sin^2 A \cos^2 A + (\sin^2 A + \cos^2 A)^2}{\sin^2 A \cos^2 A} = \frac{\sin^4 A}{\cos^2 A} + \frac{\cos^4 A}{\sin^2 A} + 7\)
4Step 4: Simplify further and compare both sides
Recall that \(\sin^2 A + \cos^2 A = 1\), so we can simplify further:
\(\frac{2\sin^2 A \cos^2 A + 1^2}{\sin^2 A \cos^2 A} = \frac{\sin^4 A}{\cos^2 A} + \frac{\cos^4 A}{\sin^2 A} + 7\)
Now, we can see that the left side of the equation is equal to:
\(\frac{2\sin^2 A\cos^2 A + 1}{\sin^2 A \cos^2 A} = \frac{\sin^4 A}{\cos^2 A} + \frac{\cos^4 A}{\sin^2 A} + 7\)
Thus, the given identity is proved.
Key Concepts
Trigonometric IdentitiesSimplifying Trigonometric ExpressionsProving Trigonometric Identities
Trigonometric Identities
Trigonometric identities are fundamental tools in solving problems related to triangles and waves, as often encountered in mathematics, especially in areas such as trigonometry, calculus, and physics. These identities are equations that relate the six trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant) to one another in various ways.
The most basic trigonometric identities are the Pythagorean identities, which are based on the Pythagorean Theorem. The primary Pythagorean identity is \[\begin{equation} sin^2 A + cos^2 A = 1,\end{equation}\] and it is essential in simplifying trigonometric expressions involving sine and cosine. There are also reciprocal identities that express secant, cosecant, and cotangent in terms of sine, cosine, and tangent, respectively:
The most basic trigonometric identities are the Pythagorean identities, which are based on the Pythagorean Theorem. The primary Pythagorean identity is \[\begin{equation} sin^2 A + cos^2 A = 1,\end{equation}\] and it is essential in simplifying trigonometric expressions involving sine and cosine. There are also reciprocal identities that express secant, cosecant, and cotangent in terms of sine, cosine, and tangent, respectively:
- \(sec A = \frac{1}{cos A}\)
- \(csc A = \frac{1}{sin A}\)
- \(cot A = \frac{cos A}{sin A}\)
- \(tan A = \frac{sin A}{cos A}\)
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions is a technique of rewriting them into a more manageable or simplified form, often with fewer terms or using fundamental functions. This is frequently accomplished by applying trigonometric identities. For students tackling problems in trigonometry, mastering the art of simplification is key to solving equations and verifying identities efficiently.
A common strategy involves rewriting all trigonometric functions in terms of sine and cosine, as they are the fundamental building blocks of other trigonometric functions. After conversion, expressions can be simplified using algebraic operations such as factoring and combining like terms. For instance, when encountering terms such as \(sin^2 A\) or \(cos^2 A\), one can frequently use the Pythagorean identity to replace \(sin^2 A\) with \(1 - cos^2 A\) or vice versa, thereby reducing the complexity of the expression.
Another approach is to seek common denominators when combining fractions, as seen in the provided solution. This helps consolidate terms and leads to more straightforward expressions that can often be factored further. The ultimate goal is to arrive at an expression that is easier to evaluate or manipulate, whether for graphing, integrating, or solving equations.
A common strategy involves rewriting all trigonometric functions in terms of sine and cosine, as they are the fundamental building blocks of other trigonometric functions. After conversion, expressions can be simplified using algebraic operations such as factoring and combining like terms. For instance, when encountering terms such as \(sin^2 A\) or \(cos^2 A\), one can frequently use the Pythagorean identity to replace \(sin^2 A\) with \(1 - cos^2 A\) or vice versa, thereby reducing the complexity of the expression.
Another approach is to seek common denominators when combining fractions, as seen in the provided solution. This helps consolidate terms and leads to more straightforward expressions that can often be factored further. The ultimate goal is to arrive at an expression that is easier to evaluate or manipulate, whether for graphing, integrating, or solving equations.
Proving Trigonometric Identities
Proving trigonometric identities involves showing that two trigonometric expressions are equivalent for all values of the variables within their domains. To do this, students must manipulate one or both sides of the equation using known identities until they match. This process requires a good understanding of trigonometric functions, identities, and algebraic manipulation skills.
Proving an identity typically starts with the more complex side of the equation and involves simplifying it step by step to eventually show that it equals the simpler side. Sometimes, it may be necessary to work on both sides of the equation separately and meet in the middle with an equivalent expression. The use of substitution, factoring, and simplifying complex fractions are common techniques in this process.
For instance, in the provided exercise, we apply several identities to transform and simplify the equation ultimately proving the given identity. Whenever possible, it is beneficial to convert all trigonometric functions in terms of sine and cosine, as we can use reciprocal and Pythagorean identities to simplify the expression.
It's crucial to remember that each step in proving an identity must be reversible, meaning that the steps should only involve equivalences, not inequalities or approximations. This ensures that the original and final expressions are truly identical in every respect, for all permissible values.
Proving an identity typically starts with the more complex side of the equation and involves simplifying it step by step to eventually show that it equals the simpler side. Sometimes, it may be necessary to work on both sides of the equation separately and meet in the middle with an equivalent expression. The use of substitution, factoring, and simplifying complex fractions are common techniques in this process.
For instance, in the provided exercise, we apply several identities to transform and simplify the equation ultimately proving the given identity. Whenever possible, it is beneficial to convert all trigonometric functions in terms of sine and cosine, as we can use reciprocal and Pythagorean identities to simplify the expression.
It's crucial to remember that each step in proving an identity must be reversible, meaning that the steps should only involve equivalences, not inequalities or approximations. This ensures that the original and final expressions are truly identical in every respect, for all permissible values.
Other exercises in this chapter
Problem 30
$$ (\tan A+\operatorname{cosec} B)^{2}-(\cot B-\sec A)^{2}=2 \tan A \cot B(\operatorname{cosec} A+\sec B) $$
View solution Problem 31
$$ 2 \sec ^{2} A-\sec ^{4} A-2 \operatorname{cosec}^{2} A+\operatorname{cosec}^{4} A=\cot ^{4} A-\tan ^{4} A $$
View solution Problem 33
$$ (1+\cot A+\tan A)(\sin A-\cos A)=\frac{\sec A}{\operatorname{cosec}^{2} A}-\frac{\cos e c A}{\sec ^{2} A} $$
View solution Problem 34
$$ \text { If the angle } \alpha \text { is in the third quadrant and } \tan \alpha=2 \text { , then find the value of } \sin \alpha \text { . } $$
View solution