Answer:
[tex]cos2\theta=\frac{7}{25}[/tex]
Step-by-step explanation:
This is a question of Trigonometric Identities. In addition to this, In quadrant IV the cosine of the angle is naturally negative. This explains the negative value for [tex]tan\theta=-3/4[/tex]
The double angle formula
Let's choose a convenient identity, for the double angle [tex]cos2\theta[/tex]
[tex]\\tan \theta=-3/4 \\ cos2\theta =cos^{2}\theta -sen^{2}\theta\\cos2\theta =2cos^{2}\theta-1\\\\1+tan\theta^{2} =sec^{2}\theta\\\\ 1+(\frac{-3}{4})^{2} =\frac{1}{cos^2 \theta} \\\\\frac{25}{16}=\frac{1}{cos^{2}\theta}\\ cos^{2}\theta=\frac{16}{25}[/tex]
Finally, we can plug it in:
[tex]cos2\theta =2cos^{2}\theta -1\\cos2\theta =2\left ( \frac{16}{25} \right )-1 \Rightarrow cos2\theta=\frac{7}{25}[/tex]
