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Chapter 4: Problem 82

Determine whether the statement is true or false. Justify your answer. $$\cot ^{2} 10^{\circ}-\csc ^{2} 10^{\circ}=-1$$

Short Answer

Expert verified
The given statement \( \cot ^{2} 10^{\circ}-\csc ^{2} 10^{\circ}=-1 \) is true.

Step by step solution

01

Express cot and csc in terms of sin and cos

From basic trigonometric identities, we have that:\(\cot \theta = \frac{\cos \theta}{\sin \theta}\) and \(\csc \theta = \frac{1}{\sin \theta}\). Let's express \( \cot ^{2} 10^{\circ} \) and \(\csc ^{2} 10^{\circ} \) in terms of sine and cosine.
02

Substitute into the given equation

Substituting the expressions from Step 1 into the given equation, we get:\(\left( \frac{\cos ^{2} 10^{\circ}}{\sin ^{2} 10^{\circ}} \right) - \left( \frac{1}{\sin ^{2} 10^{\circ}} \right) = -1\).
03

Simplify the expression

We can simplify the above equation by making the denominators equal:\(\frac{\cos ^{2} 10^{\circ} - 1}{\sin ^{2} 10^{\circ}} = -1\).
04

Use Pythagorean identity

The Pythagorean identity in trigonometry states that \( \sin^2 \theta + \cos^2 \theta = 1 \). Therefore, \( \cos^2 \theta - 1 = - \sin^2 \theta \). Substituting this into our equation, we get:\(-1 = -1\).
05

Verify the result

The equation from step 4 is true, thus the initial statement \( \cot ^{2} 10^{\circ}-\csc ^{2} 10^{\circ}=-1 \) is also true.

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