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Chapter 5: Problem 15

Verify each identity. $$\sin ^{2} \theta\left(1+\cot ^{2} \theta\right)=1$$

Short Answer

Expert verified
Verification of the given trigonometric identity has been accomplished successfully as the equation \( \sin ^{2} \theta (1+\cot ^{2} \theta) = 1 \) has been proven.

Step by step solution

01

Start with the left side

We begin with the left-hand side (LHS) of the equation: \(LHS = \sin ^{2} \theta (1+\cot ^{2} \theta) \)
02

Apply the fact that \( \cot \theta = \frac{1}{\tan \theta} \)

We can rewrite the identity \(\cot \theta\) as \(\frac{1}{\tan \theta}\), and since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), the cotangent expression \(\cot ^{2} \theta\) can be written as \(\frac{1}{\tan ^{2} \theta} = \frac{\cos ^{2} \theta}{\sin ^{2} \theta} \). Now the expression becomes \(\sin ^{2} \theta (1+ \frac{\cos ^{2} \theta}{\sin ^{2} \theta})\)
03

Simplify the expression

The term inside brackets can be further simplified by turning it into a common fraction. Therefore, we get \(LHS = \sin ^{2} \theta + \cos ^{2} \theta \)
04

Use the identity \( \sin ^{2} \theta + \cos ^{2} \theta = 1 \)

Replacing \(\sin ^{2} \theta + \cos ^{2} \theta\) with \(1\), due to the trigonometric identity, gives us the right-hand side (RHS) of the original equation: \(LHS = RHS = 1\)

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