Use identities to fill in the blank to complete the proof. Use "theta" for [tex]$\theta$[/tex].

[tex]\cos(\theta) \cdot \tan(\theta) = \cos(\theta) \cdot \frac{\sin(\theta)}{\cos(\theta)} = \sin(\theta)[/tex]



Answer :

Let's complete the proof using trigonometric identities.

We need to show that:

[tex]\[ \cos (\theta) \cdot \tan (\theta) = \sin (\theta) \][/tex]

First, recall the definition of the tangent function in terms of sine and cosine:

[tex]\[ \tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)} \][/tex]

Next, substitute this expression for [tex]\(\tan (\theta)\)[/tex] into the left-hand side of the equation:

[tex]\[ \cos (\theta) \cdot \tan (\theta) = \cos (\theta) \cdot \left( \frac{\sin (\theta)}{\cos (\theta)} \right) \][/tex]

We can simplify this by canceling [tex]\(\cos (\theta)\)[/tex] from the numerator and the denominator:

[tex]\[ \cos (\theta) \cdot \left( \frac{\sin (\theta)}{\cos (\theta)} \right) = \sin (\theta) \][/tex]

Therefore, we have shown that:

[tex]\[ \cos (\theta) \cdot \tan (\theta) = \sin (\theta) \][/tex]

And this completes the proof.