Answer:
[tex]-\dfrac{7}{25}[/tex]
Step-by-step explanation:
Trigonometric Identities
[tex]\cos(A \pm B)=\cos A \cos B \mp \sin A \sin B[/tex]
Trigonometric ratios
[tex]\sf \sin(\theta)=\dfrac{O}{H}\quad\cos(\theta)=\dfrac{A}{H}\quad\tan(\theta)=\dfrac{O}{A}[/tex]
where:
- [tex]\theta[/tex] is the angle
- O is the side opposite the angle
- A is the side adjacent the angle
- H is the hypotenuse (the side opposite the right angle)
Using the trig ratio formulas for cosine and sine:
- [tex]\cos(\angle ABC)=\dfrac{3}{5}[/tex]
- [tex]\sin(\angle ABC)=\dfrac{4}{5}[/tex]
Therefore, using the trig identities and ratios:
[tex]\begin{aligned}\implies \cos(2 \cdot \angle ABC) & = \cos(\angle ABC + \angle ABC)\\\\& = \cos (\angle ABC) \cos (\angle ABC) - \sin(\angle ABC) \sin (\angle ABC)\\\\& = \cos^2(\angle ABC)-\sin^2(\angle ABC)\\\\& = \left(\dfrac{3}{5}\right)^2-\left(\dfrac{4}{5}\right)^2\\\\& = \dfrac{3^2}{5^2}-\dfrac{4^2}{5^2}\\\\& = \dfrac{9}{25}-\dfrac{16}{25}\\\\& = \dfrac{9-16}{25}\\\\& = -\dfrac{7}{25} \end{aligned}[/tex]
