Answer :
Answer:
[tex]csc\theta=\frac{5}{3}[/tex]
Step-by-step explanation:
Given:
[tex]tan\theta =\frac{3}{4}[/tex]
[tex]cot\theta =\frac{4}{3}[/tex] [∵ [tex]cot\theta =\frac{1}{tan\theta}[/tex] ]
Squaring both sides.
[tex]cot^2\theta =\frac{4^2}{3^2}=\frac{16}{9}[/tex]
[tex]csc^2\theta -1=\frac{16}{9}[/tex] [∵ [tex]cot^2\theta =csc^2\theta-1][/tex] ]
Adding 1 to both sides.
[tex]csc^2\theta -1+1=\frac{16}{9}+1[/tex]
[tex]csc^2\theta =\frac{16}{9}+1[/tex]
[tex]csc^2\theta=\frac{16}{9} +\frac{9}{9} [/tex] [Taking LCD=9 and adding fractions ]
[tex]csc^2\theta=\frac{16+9}{9}[/tex]
[tex]csc^2\theta=\frac{25}{9}[/tex]
Taking square root both sides.
[tex]\sqrt{csc^2\theta}=\sqrt{\frac{25}{9}}[/tex]
∴ [tex]csc\theta=\frac{5}{3}[/tex]