PROBLEM STATEMENT
To evaluate the value of
[tex]\tan (\alpha+\beta)[/tex]GIVEN
[tex]\begin{gathered} \cos (\alpha+\beta)=-\frac{528}{697} \\ \sin (\alpha+\beta)=\frac{455}{697} \\ \sin (\alpha)=\frac{40}{41} \\ \sin (\beta)=\frac{15}{17} \end{gathered}[/tex]SOLUTION
Recall the trigonometric identity:
[tex]\tan (x)=\frac{\sin (x)}{\cos (x)}[/tex]If we have
[tex]x=\alpha+\beta[/tex]Therefore, we have that:
[tex]\tan (\alpha+\beta)=\frac{\sin(\alpha+\beta)}{\cos(\alpha+\beta)}[/tex]Substituting for the values of sin and cos, we have:
[tex]\tan (\alpha+\beta)=\frac{\frac{455}{697}}{-\frac{528}{697}}[/tex]Rewriting, we have:
[tex]\begin{gathered} \tan (\alpha+\beta)=-\frac{455}{697}\div\frac{528}{697} \\ \tan (\alpha+\beta)=-\frac{455}{697}\times\frac{697}{528} \\ \tan (\alpha+\beta)=-\frac{455}{528} \end{gathered}[/tex]ANSWER
[tex]\tan (\alpha+\beta)=-\frac{455}{528}[/tex]