Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Get immediate and reliable solutions to your questions from a community of experienced experts on our Q&A platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To provide a step-by-step explanation and review of the proof, we need to focus on how to transition from step 1 to step 2:
[tex]\[ \begin{array}{|l|l|} \hline \text{Step} & \text{Statement} \\ \hline 1 & \tan \left(\frac{x}{2}\right)=\sqrt{\frac{1-\cos (x)}{1+\cos (x)}} \\ \hline 2 & \tan \left(\frac{x}{2}\right)=\left(\frac{\sqrt{1-\cos (x)}}{\sqrt{1+\cos (x)}}\right)\left(\frac{7}{\sqrt{1+\cos (x)}}\right) \\ \hline \end{array} \][/tex]
In step 1, we have:
[tex]\[ \tan\left(\frac{x}{2}\right) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}} \][/tex]
Step 2 shows:
[tex]\[ \tan \left(\frac{x}{2}\right)=\left(\frac{\sqrt{1-\cos (x)}}{\sqrt{1+\cos (x)}}\right)\left(\frac{7}{\sqrt{1+\cos (x)}}\right) \][/tex]
To complete the expression in step 2, we notice that the fraction inside the square root in step 1:
[tex]\[ \sqrt{\frac{1-\cos(x)}{1+\cos(x)}} \][/tex]
is split into two parts:
[tex]\[ \left(\frac{\sqrt{1-\cos (x)}}{\sqrt{1+\cos (x)}}\right) \][/tex]
and another term:
[tex]\[ \left(\frac{7}{\sqrt{1+\cos (x)}}\right) \][/tex]
To transition from step 1 to step 2, we multiply and divide by [tex]\(\sqrt{1+\cos(x)}\)[/tex]. Thus, we are effectively introducing the denominator [tex]\(\sqrt{1+\cos(x)}\)[/tex] again in both the numerator and denominator of the fraction.
The expression that completes the step is:
[tex]\[ 1 + \cos(x) \][/tex]
When we multiply the numerator and denominator in the fraction [tex]\(\sqrt{\frac{1-\cos(x)}{1+\cos(x)}}\)[/tex] by [tex]\(\sqrt{1+\cos(x)}\)[/tex], we obtain (after splitting appropriately):
[tex]\[ \tan \left(\frac{x}{2}\right) = \left( \frac{\sqrt{1-\cos(x)}}{\sqrt{1+\cos(x)}} \right) \left( \frac{\sqrt{1+\cos(x)}}{\sqrt{1+\cos(x)}} \right) \][/tex]
Thus, incorporating the given expression correctly aligns the proof.
Therefore, the expression that will complete step 2 in the proof is:
[tex]\[ 1 + \cos (x) \][/tex]
[tex]\[ \begin{array}{|l|l|} \hline \text{Step} & \text{Statement} \\ \hline 1 & \tan \left(\frac{x}{2}\right)=\sqrt{\frac{1-\cos (x)}{1+\cos (x)}} \\ \hline 2 & \tan \left(\frac{x}{2}\right)=\left(\frac{\sqrt{1-\cos (x)}}{\sqrt{1+\cos (x)}}\right)\left(\frac{7}{\sqrt{1+\cos (x)}}\right) \\ \hline \end{array} \][/tex]
In step 1, we have:
[tex]\[ \tan\left(\frac{x}{2}\right) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}} \][/tex]
Step 2 shows:
[tex]\[ \tan \left(\frac{x}{2}\right)=\left(\frac{\sqrt{1-\cos (x)}}{\sqrt{1+\cos (x)}}\right)\left(\frac{7}{\sqrt{1+\cos (x)}}\right) \][/tex]
To complete the expression in step 2, we notice that the fraction inside the square root in step 1:
[tex]\[ \sqrt{\frac{1-\cos(x)}{1+\cos(x)}} \][/tex]
is split into two parts:
[tex]\[ \left(\frac{\sqrt{1-\cos (x)}}{\sqrt{1+\cos (x)}}\right) \][/tex]
and another term:
[tex]\[ \left(\frac{7}{\sqrt{1+\cos (x)}}\right) \][/tex]
To transition from step 1 to step 2, we multiply and divide by [tex]\(\sqrt{1+\cos(x)}\)[/tex]. Thus, we are effectively introducing the denominator [tex]\(\sqrt{1+\cos(x)}\)[/tex] again in both the numerator and denominator of the fraction.
The expression that completes the step is:
[tex]\[ 1 + \cos(x) \][/tex]
When we multiply the numerator and denominator in the fraction [tex]\(\sqrt{\frac{1-\cos(x)}{1+\cos(x)}}\)[/tex] by [tex]\(\sqrt{1+\cos(x)}\)[/tex], we obtain (after splitting appropriately):
[tex]\[ \tan \left(\frac{x}{2}\right) = \left( \frac{\sqrt{1-\cos(x)}}{\sqrt{1+\cos(x)}} \right) \left( \frac{\sqrt{1+\cos(x)}}{\sqrt{1+\cos(x)}} \right) \][/tex]
Thus, incorporating the given expression correctly aligns the proof.
Therefore, the expression that will complete step 2 in the proof is:
[tex]\[ 1 + \cos (x) \][/tex]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.