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Chapter 29: Problem 961

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 trigonometric functions in terms of sine and cosine, and simplifying both sides of the equation. We find that \(\frac{1}{\cos A \cdot \sin A} = \frac{\sin^2 A + \cos^2 A}{\cos A \cdot \sin A}\). Using the Pythagorean identity \(\sin^2 A + \cos^2 A = 1\), we confirm that both sides of the equation are equal, proving the identity.

Step by step solution

01

Define and rewrite trigonometric functions

Start by defining the trigonometric functions involved. The trigonometric functions are defined as follows: \[sec(A) = \frac{1}{cos(A)}\] \[csc(A) = \frac{1}{sin(A)}\] \[tan(A) = \frac{sin(A)}{cos(A)}\] \[cot(A) = \frac{cos(A)}{sin(A)}\] Now, we need to rewrite the given identity using the defined trigonometric functions in terms of sine and cosine: \[\sec A \csc A=\tan A+\cot A\] \[\frac{1}{\cos A} \cdot \frac{1}{\sin A} = \frac{\sin A}{\cos A} + \frac{\cos A}{\sin A}\]
02

Simplify the left-hand side of the identity

We can simplify the left-hand side of the given identity by multiplying the fractions on the left side \[\frac{1}{\cos A \cdot \sin A} = \frac{\sin A}{\cos A} + \frac{\cos A}{\sin A}\]
03

Simplify the right-hand side of the identity

To make the denominators the same on the right-hand side of the identity, we will find a common denominator, which is the product of both denominators, i.e., \(\cos A \cdot \sin A\). Then we rewrite the right-hand side: \[\frac{1}{\cos A \cdot \sin A} = \frac{\sin^2 A + \cos^2 A}{\cos A \cdot \sin A}\]
04

Use the Pythagorean identity

We know that one of the fundamental Pythagorean identities is \[\sin^2 A + \cos^2 A = 1\] Substitute this identity into the right-hand side of the equation: \[\frac{1}{\cos A \cdot \sin A} = \frac{1}{\cos A \cdot \sin A}\] As both sides of the equation are equal, we have now shown that \[\sec A \csc A = \tan A + \cot A\] which means the identity is proved.

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Most popular questions from this chapter

Find the solution set on \([0,2 \pi]\) of the equation \(\left(\sqrt{1}+\sin ^{2} x\right)=(\sqrt{2}) \sin x\).

Prove that \(1 /(\sec A-\tan A)=\sec A+\tan A\) is an identity.

Graph \(\mathrm{y}=\csc \mathrm{x}, 0 \leq \mathrm{x} \leq 2 \pi\).

Prove the identity \(\left(\sin ^{2} \theta+\cos ^{2} \theta\right) / \cos ^{2} \theta=\sec ^{2} \theta\).

If \(\mathrm{u}\) and \(\mathrm{v}\) are two numbers such that \(\mathrm{u}+\mathrm{v}=1 / 2 \pi\), show that \(\sin ^{2} u+\sin ^{2} v=1\)
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