With [tex]\theta[/tex] in quadrant 3, we should expect both [tex]\cos\theta[/tex] and [tex]\sin\theta[/tex] to be negative, so that [tex]\tan\theta[/tex] is positive. The corresponding reciprocal expressions [tex](\sec\theta,\csc\theta,\cot\theta)[/tex] will have the same sign.
[tex]\cot\theta=\dfrac34\implies\tan\theta=\dfrac43[/tex]
Recall that [tex]1+\tan^2\theta=\sec^2\theta[/tex], which means
[tex]\sec\theta=-\sqrt{1+\tan^2\theta}=-\dfrac53[/tex]
[tex]\implies\cos\theta=-\dfrac35[/tex]
Also recall that [tex]\cos^2\theta+\sin^2\theta=1[/tex], so
[tex]\sin\theta=-\sqrt{1-\cos^2\theta}=-\dfrac45[/tex]
[tex]\implies\csc\theta=-\dfrac54[/tex]
Only the first two options are correct.
