If [tex]\theta[/tex] falls in quadrant IV, then we know [tex]\sin\theta<0[/tex] and [tex]\cos\theta>0[/tex]. By definition of cosecant,
[tex]\csc\theta=\dfrac1{\sin\theta}[/tex]
so we also know that [tex]\csc\theta<0[/tex]. Recall that
[tex]\cot^2\theta+1=\csc^2\theta[/tex]
which means
[tex]\csc\theta=-\sqrt{\cot^2\theta+1}=-\dfrac{\sqrt{10}}3[/tex]
[tex]\implies\sin\theta=-\dfrac3{\sqrt{10}}[/tex]
By definition of cotangent,
[tex]\cot\theta=\dfrac{\cos\theta}{\sin\theta}\implies\cos\theta=\dfrac1{\sqrt{10}}[/tex]
[tex]\implies\sec\theta=\sqrt{10}[/tex]
We also immediately know that
[tex]\tan\theta=-\dfrac{21}7[/tex]
The listed answers are unsimplified relative to the ones we've come up with here, but with some manipulation we find
[tex]\sin\theta=-\dfrac3{\sqrt{10}}=-\dfrac{7\cdot3}{7\sqrt{10}}=-\dfrac{21}{\sqrt{490}}[/tex]
[tex]\cos\theta=\dfrac1{\sqrt{10}}=\dfrac7{7\sqrt{10}}=\dfrac7{\sqrt{490}}[/tex]
[tex]\csc\theta=\dfrac1{\sin\theta}=-\dfrac{\sqrt{490}}{21}[/tex]
[tex]\sec\theta=\dfrac1{\cos\theta}=\dfrac{\sqrt{490}}7[/tex]
[tex]\tan\theta=\dfrac1{\cot\theta}=-\dfrac{21}7[/tex]
so that the third option is correct.
