Answered

\begin{tabular}{|l|c|}
\hline
Statement & Reason \\
\hline
[tex]$\cos (2 x)$[/tex] & Given \\
\hline
[tex]$=\cos (x+x)$[/tex] & 1 \\
\hline
[tex]$=\cos (x) \cos (x)-\sin (x) \sin (x)$[/tex] & 2 \\
\hline
[tex]$=\cos ^2(x)-\sin ^2(x)$[/tex] & 3 \\
\hline
\end{tabular}

The table shows the derivation of [tex]$\cos (2 x)$[/tex]. Select the correct reason for each step.

Reason 1 is [tex]$\square$[/tex] (Angle Addition Formula)

Reason 2 is [tex]$\square$[/tex] (Cosine of a Sum Formula)

Reason 3 is [tex]$\square$[/tex] (Simplification)



Answer :

GinnyAnswer
Let's go through each step and match it with the correct reason:

1. Statement: [tex]\( \cos(2x) = \cos(x + x) \)[/tex]
- Reason: This step involves using the angle addition formula for cosine. Thus, Reason 1 is: "Use angle addition formula for cosine".

2. Statement: [tex]\( \cos(x + x) = \cos(x)\cos(x) - \sin(x)\sin(x) \)[/tex]
- Reason: This step involves applying the specific cosine addition formula, [tex]\(\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B)\)[/tex]. Thus, Reason 2 is: "Apply the formula [tex]\(\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B)\)[/tex]".

3. Statement: [tex]\( \cos(x)\cos(x) - \sin(x)\sin(x) = \cos^2(x) - \sin^2(x) \)[/tex]
- Reason: This step involves simplifying the expression by combining like terms. Thus, Reason 3 is: "Simplify the expression".

So, the final reasoning for each step is as follows:

Reason 1 is: Use angle addition formula for cosine

Reason 2 is: Apply the formula [tex]\(\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B)\)[/tex]

Reason 3 is: Simplify the expression

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