Question

verify the identity.
( \tan \theta cdot cos \theta cdot csc \theta = 1 )
which of the following four statements establishes the identity?
a. ( \tan \theta cdot cos \theta cdot csc \theta=\frac{sin \theta}{cos \theta} cdot cos \theta cdot \frac{1}{sin \theta}=1 )
b. ( \tan \theta cdot cos \theta cdot csc \theta=\frac{cos \theta}{sin \theta} cdot cos \theta cdot \frac{1}{cos \theta}=1 )
c. ( \tan \theta cdot cos \theta cdot csc \theta=\frac{sin \theta}{cos \theta} cdot cos \theta cdot \frac{1}{cos \theta}=1 )
d. ( \tan \theta cdot cos \theta cdot csc \theta=\frac{cos \theta}{sin \theta} cdot cos \theta cdot \frac{1}{sin \theta}=1 )

Explanation:

Step1: Recall trigonometric identities

Recall that \(\tan\theta=\frac{\sin\theta}{\cos\theta}\) and \(\csc\theta = \frac{1}{\sin\theta}\).

Step2: Substitute identities into the left - hand side

Substitute \(\tan\theta=\frac{\sin\theta}{\cos\theta}\) and \(\csc\theta=\frac{1}{\sin\theta}\) into \(\tan\theta\cdot\cos\theta\cdot\csc\theta\).
We get \(\frac{\sin\theta}{\cos\theta}\cdot\cos\theta\cdot\frac{1}{\sin\theta}\).

Step3: Simplify the expression

Cancel out \(\sin\theta\) and \(\cos\theta\) terms. \(\frac{\sin\theta}{\cos\theta}\cdot\cos\theta\cdot\frac{1}{\sin\theta}=\frac{\sin\theta\times\cos\theta\times1}{\cos\theta\times\sin\theta}=1\)

Answer:

A. \(\tan\theta\cdot\cos\theta\cdot\csc\theta=\frac{\sin\theta}{\cos\theta}\cdot\cos\theta\cdot\frac{1}{\sin\theta}=1\)