Recall that
[tex]\cos^2\theta+\sin^2\theta=1\implies\cos\theta=\pm\sqrt{1-\sin^2\theta}[/tex]
But we're specifically told that cosine is negative, so we take the negative root above, and then
[tex]\cos\theta=-\sqrt{1-\left(\dfrac34\right)^2}=-\dfrac{\sqrt7}4[/tex]
Now,
[tex]\csc\theta=\dfrac1{\sin\theta}=\dfrac43[/tex]
[tex]\cot\theta=\dfrac{\cos\theta}{\sin\theta}=-\dfrac{\sqrt7}3[/tex]
