Problem 771 Prove the identity: \(\sec A \cs... [FREE SOLUTION]

Chapter 35: Problem 771

Prove the identity: \(\sec A \csc A=\tan A+\cot A\).

Short Answer

Expert verified

We can prove the identity \(\sec A \csc A = \tan A + \cot A\) by rewriting the given expression using the definitions of secant, cosecant, tangent, and cotangent in terms of sine and cosine: \(\frac{1}{\cos A} \cdot \frac{1}{\sin A} = \frac{\sin A}{\cos A} + \frac{\cos A}{\sin A}\). We then find a common denominator by multiplying both sides of the equation by \(\sin A\cos A\), which results in the equation: \(\frac{\sin A\cos A}{\sin A\cos A} = \frac{\sin^2 A+\cos^2 A}{\sin A\cos A}\). This simplifies to \(\frac{1}{\sin A\cos A} = \frac{1}{\sin A\cos A}\), proving the given identity.

Step by step solution

Rewrite the given identity using the trigonometric functions

Replace \(\sec A\), \(\csc A\), \(\tan A\), and \(\cot A\) with their definitions using sine and cosine functions: \[\frac{1}{\cos A} \cdot \frac{1}{\sin A} = \frac{\sin A}{\cos A} + \frac{\cos A}{\sin A}\]

Find a common denominator

Since sine and cosine are common denominators, we will multiply both sides of the equation by sin(A)cos(A) to create a common denominator for the terms: \[(\frac{1}{\cos A} \cdot \frac{1}{\sin A})\cdot (\sin A\cos A) = (\frac{\sin A}{\cos A} + \frac{\cos A}{\sin A})\cdot (\sin A\cos A)\]

Simplify both sides

Now we need to simplify both sides of the equation by cancelling out common factors: \[\frac{\sin A\cos A}{\sin A\cos A} = \frac{\sin^2 A+\cos^2 A}{\sin A\cos A}\] The left side of the equation simplifies to 1, and since \(\sin^2 A+\cos^2 A=1\), the right side of the equation is: \[\frac{1}{\sin A\cos A} = \frac{1}{\sin A\cos A}\]

Conclude the proof

We have shown that both sides of the equation are equal, so we have proven the given identity: \[\sec A \csc A = \tan A + \cot A\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Secant

The secant function, \( \sec A \), is a fundamental trigonometric identity. It is the reciprocal of the cosine function. In simpler words, if you know the cosine of an angle, you can find the secant by taking the inverse. \

Definition: \( \sec A = \frac{1}{\cos A} \) \
Secant is undefined where cosine is zero; these points are at \( 90^\circ, 270^\circ, \cdots \). \
It is used in proofs, often to simplify complex trigonometric expressions by linking angles to their respective ratios. \

\Secant can be useful when proving identities or solving equations, as seen in the exercise above. By isolating secant, other trigonometric forms can reveal themselves, aiding in the simplification process.

Cosecant

Cosecant, \( \csc A \), is another reciprocal trigonometric function, complementary to the sine function. It represents how many times the sine value fits into one. \

Definition: \( \csc A = \frac{1}{\sin A} \) \
Undefined Points: Where the sine value is zero, the cosecant is undefined (e.g., \( 0^\circ, 180^\circ, \cdots \)). \
Application: Cosecant is crucial in transforming expressions, particularly when paired with other reciprocals like secant. \

\In exercises like this one, knowing the cosecant helps in equalizing both sides of the equation through expression matching and cancellation.

Tangent

Tangent, denoted as \( \tan A \), connects the sine and cosine functions. Essentially, it is the ratio of sine to cosine for a specific angle, giving us insight into the angular geometry. \

Definition: \( \tan A = \frac{\sin A}{\cos A} \) \
Properties: Tangent is zero where sine is zero, as long as cosine isn't also zero. This relationship provides angles such as \( 0^\circ, 180^\circ, \cdots \). \
Usage: In trigonometric proofs, \( \tan A \) often simplifies the problem as it cancels out parts of expressions. \

\Tangent often works together with secant and cosecant to achieve balance in expressions. By observing tangent in its reciprocal or original form, complex identities become approachable.

Cotangent

Cotangent, symbolized as \( \cot A \), complements tangent by flipping it over, so it's the reciprocal of the tangent function. Here's how it fits in with other trigonometric terms:

Definition: \( \cot A = \frac{\cos A}{\sin A} \)
Zero Points: Cotangent becomes undefined where sine is zero because these are poles, not zeros for cotangent.
Function: Cotangent supplements proofs by allowing equivalence demonstrations, working especially well in paired functions with tangent.

In this exercise, \( \cot A \) is crucial as it balances \( \tan A \) on the other side of the equation. Such relationships help in solving and proving intricate trigonometric equations efficiently.

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91影视

Prove the identity: \(\sec A \csc A=\tan A+\cot A\).

Short Answer

Step by step solution

Rewrite the given identity using the trigonometric functions

Find a common denominator

Simplify both sides

Conclude the proof

Key Concepts

Secant

Cosecant

Tangent

Cotangent

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