Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Dec 07, 2023

Prove the trigonometric identity: $\cos \theta \cdot \csc \theta=\cot \theta$ .

STEP 1

Assumptions
1. We are working with trigonometric identities.
2. The variable $\theta$ represents an angle in a right triangle or in the unit circle.
3. The trigonometric functions involved are cosine ( $\cos$ ), cosecant ( $\csc$ ), and cotangent ( $\cot$ ).
4. The identity we want to establish is $\cos \theta \cdot \csc \theta = \cot \theta$ .

STEP 2

Recall the definitions of the trigonometric functions in terms of the sides of a right triangle. For an angle $\theta$ in a right triangle with hypotenuse $h$ , opposite side $o$ , and adjacent side $a$ :
$\cos \theta = \frac{a}{h}$ $\csc \theta = \frac{1}{\sin \theta} = \frac{h}{o}$ $\cot \theta = \frac{1}{\tan \theta} = \frac{a}{o}$

STEP 3

Write the left side of the identity using these definitions:
$\cos \theta \cdot \csc \theta = \left(\frac{a}{h}\right) \cdot \left(\frac{h}{o}\right)$

STEP 4

Simplify the expression by canceling the common factor $h$ :
$\cos \theta \cdot \csc \theta = \frac{a}{h} \cdot \frac{h}{o} = \frac{a \cdot h}{h \cdot o} = \frac{a}{o}$

STEP 5

Recognize that the simplified expression $\frac{a}{o}$ is the definition of $\cot \theta$ :
$\cot \theta = \frac{a}{o}$

STEP 6

Conclude that the left side of the identity is equal to the right side:
$\cos \theta \cdot \csc \theta = \cot \theta$
Therefore, the identity is established.

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