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Chapter 5: Problem 72

Determine whether the equation is an identity, and give a reason for your answer. $$\sin \theta \csc \theta=1$$

Short Answer

Expert verified
The given equation \( \sin \theta \csc \theta = 1 \) is an identity since it is true for all possible values of \( \theta \).

Step by step solution

01

Define the trigonometric functions

The cosecant function, \( \csc \theta \), is defined as \( 1/\sin \theta \). Thus, we can replace \( \csc \theta \) with \( 1/\sin \theta \) in the given equation.
02

Simplify the equation

Replacing \( \csc \theta \) with \( 1/\sin \theta \), the equation \( \sin \theta \csc \theta = 1 \) becomes \( \sin \theta *(1/\sin \theta) = 1 \). Simplifying this, we get \( 1 = 1 \) since \( \sin \theta \) cancels out.
03

Determine identity

The equation \( 1 = 1 \) is true for all possible values of \( \theta \), which confirms that the given equation is an identity.

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