One is given the following:
- [tex]cos(2A)=\frac{11}{25}[/tex]
One is asked to prove the following:
- [tex]cos(A)=\frac{6}{5\sqrt{2}}[/tex]
In order to prove the statement above, one will need to use a trigonometric identity. In this case, the following identity is the most relevant in the proof.
[tex]cos(2a)=2cos^2(A)-1[/tex]
One can manipulate this identity to suit the needs of the given problem:
[tex]cos(2a)=2cos^2(A)-1[/tex]
[tex]cos(2a)+1=2cos^2(A)[/tex]
[tex]\frac{cos(2a)+1}{2}=cos^2(A)[/tex]
[tex]\sqrt{\frac{cos(2a)+1}{2}}=cos(A)[/tex]
Now substitute the given information into this identity,
[tex]cos(2A)=\frac{11}{25}[/tex]
[tex]\sqrt{\frac{cos(2a)+1}{2}}=cos(A)[/tex]
Substitute,
[tex]\sqrt{\frac{\frac{11}{25}+1}{2}}=cos(A)[/tex]
Simplify, remember, any number over itself equals (1) and, in order to add two fractions, they must have the same denominator.
[tex]\sqrt{\frac{\frac{11}{25}+1}{2}}=cos(A)[/tex]
[tex]\sqrt{\frac{\frac{11}{25}+\frac{25}{25}}{2}}=cos(A)[/tex]
[tex]\sqrt{\frac{\frac{36}{25}}{2}}=cos(A)[/tex]
[tex]\sqrt{\frac{18}{25}}=cos(A)[/tex]
[tex]\frac{3\sqrt{2}}{5}=cos(A)[/tex]
Manipulate so that it resembles the given information; remember, any number over itself is (1), multiplying an equation by (1) doesn't change it,
[tex]\frac{3\sqrt{2}}{5}=cos(A)[/tex]
[tex]\frac{3\sqrt{2}}{5}*\frac{\sqrt{2}}{\sqrt{2}}=cos(A)[/tex]
[tex]\frac{3\sqrt{2*2}}{5*\sqrt{2}}=cos(A)[/tex]
[tex]\frac{6}{5\sqrt{2}}=cos(A)[/tex]
