We are given four expressions and we want to know which one of them is not equivalent to cosθ.
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Expression A. & - sin (θ-90^(∘)) [0.8em]
Expression B. & - cos (- θ) [0.8em]
Expression C. & sin (θ + 90^(∘)) [0.8em]
Expression D. &- cos (θ + 180^(∘))
To do so, let's simplify one expression at a time.
Expression A
We can start by rewriting the parenthetical expression of Expression A. We will factor out - 1, and rewrite 90 ^(∘) as π2.
Now, we will recall one of the Cofunction Identities.
sin ( π/2 - θ ) = cos θ
Therefore, Expression A is equal to cos θ.
Expression B
Recall the Negative Angle Identity for cosine.
cos (- θ) = cos θ
We will simplify Expression B by applying this identity. Let's do it!
-cos (- θ) ⇒ - cos θ
As we can see, the Negative Angle Identity changed the sign of the expression inside the parentheses but the sign of the whole expression remains negative. Therefore, this expression is not equivalent to cosθ.
Expression C
For Expression C, we can use the Angle Sum Identity for sine.
sin ( A + B) = sin A cos B + cos A sin B
Now, let's apply this identity to the given expression.
Finally, let's remember the Angle Sum Identity for cosine.
cos ( A + B) = cos A cos B - sin A sin B
We can apply this identity to simplify Expression D.
