if $csc\theta = \frac{8}{7}$, which equation represents $cot\theta$?\n$cot\theta=\frac{sqrt{15}}{8}$\n$cot\th…

Explanation:

Step1: Recall the reciprocal relationship

Since $\csc\theta=\frac{8}{7}$, then $\sin\theta = \frac{7}{8}$ (because $\sin\theta=\frac{1}{\csc\theta}$).

Step2: Use the Pythagorean identity

We know that $\sin^{2}\theta+\cos^{2}\theta = 1$. Substitute $\sin\theta=\frac{7}{8}$ into it: $\left(\frac{7}{8}\right)^{2}+\cos^{2}\theta=1$. Then $\cos^{2}\theta=1 - \frac{49}{64}=\frac{64 - 49}{64}=\frac{15}{64}$, so $\cos\theta=\pm\frac{\sqrt{15}}{8}$.

Step3: Recall the cotangent - sine - cosine relationship

Since $\cot\theta=\frac{\cos\theta}{\sin\theta}$, substituting $\sin\theta = \frac{7}{8}$ and $\cos\theta=\frac{\sqrt{15}}{8}$ (assuming we are in the first - quadrant where all trigonometric functions are positive for simplicity, the sign doesn't affect the magnitude of the ratio for this problem), we get $\cot\theta=\frac{\frac{\sqrt{15}}{8}}{\frac{7}{8}}=\frac{\sqrt{15}}{7}$.

Answer:

$\cot\theta=\frac{\sqrt{15}}{7}$