Answer
Step 1. $cos\theta=|OP|$
Step 2. $tan\theta=|RQ|$
Step 3. $csc\theta=|OS|$
Step 4. $sec\theta=|OQ|$
Step 5. $cot\theta=|RS|$
Work Step by Step
Step 1. Use the figure provided with the Exercise, we have $cos\theta=\frac{|OP|}{|OR|}=|OP|$ (since
|OR|=1)
Step 2. $tan\theta=\frac{|PR|}{|OP|}$ since $\Delta OPR$ and $\Delta ORQ$ are similar triangle, we have
$\frac{PR}{OP}=\frac{RQ}{OR}=RQ$, we have $tan\theta=|RQ|$
Step 3. $csc\theta=\frac{1}{sin\theta}=\frac{1}{|PR|}$, use the properties of similar triangles of $\Delta OPR$ and $\Delta ORS$, we have $\frac{PR}{OR}=\frac{OR}{OS}$ so that $\frac{1}{|PR|}=|OS|$ which means that $csc\theta=|OS|$
Step 4. $sec\theta=\frac{1}{cos\theta}=\frac{1}{|OP|}$, use the properties of similar triangles of $\Delta OPR$ and $\Delta ORQ$, we have $\frac{OP}{OR}=\frac{OR}{OQ}$ so that $\frac{1}{|OP|}=|OQ|$ which means that $sec\theta=|OQ|$
Step 5. $cot\theta=\frac{|OP|}{|PR|}$ since $\Delta OPR$ and $\Delta ORS$ are similar triangle, we have
$\frac{OP}{PR}=\frac{RS}{OR}=RS$, we have $cot\theta=|RS|$
