Cos 2 A = [tex]\frac{7}{25}[/tex] for the given.
Step-by-step explanation:
Given:
[tex]\cos A=-\frac{4}{5}[/tex]
We know the formula,
[tex]\cos ^{2} A+\sin ^{2} A=1[/tex]
[tex]\sin ^{2} A=1-\cos ^{2} A[/tex]
[tex]\sin ^{2} A=1-\left(-\frac{4}{5}\right)^{2}=1-\frac{16}{25}=\frac{25-16}{25}=\frac{9}{25}[/tex]
Taking square root, we get
[tex]\sin A=\sqrt{\frac{9}{25}}=\pm \frac{3}{5}[/tex]
Hence,
[tex]\sin A=+\frac{3}{5} \text { and } \sin A=-\frac{3}{5}[/tex]
Given as ‘A’ is in second quadrant, so sine value is positive. Therefore,
[tex]\sin A=+\frac{3}{5}[/tex]
The formula for Cos 2 A is given as
[tex]\cos 2 A=\cos ^{2} A-\sin ^{2} A[/tex]
Substitute the values, we get
[tex]\cos 2 A=\left(-\frac{4}{5}\right)^{2}-\left(+\frac{3}{5}\right)^{2}[/tex]
[tex]\cos 2 A=\frac{16}{25}-\frac{9}{25}=\frac{16-9}{25}=\frac{7}{25}[/tex]
