In case it is not clear from where some of the formulas in the leftmost column of the table came, we want to show how these equations were derived. Let's first derive the formula for sin θ. We will use basic identities to show how the right-hand side matches the left-hand side.
\sin \theta \stackrel{?}= \dfrac{\tan \theta}{\sec \theta}
\sin \theta \stackrel{?}= \tan \theta\left(\dfrac{1}{\sec \theta}\right)
\sin \theta \stackrel{?}= {\color{#0000FF}{\dfrac{\sin \theta}{\cos \theta}}}\left(\dfrac{1}{\sec \theta}\right)
\sin \theta \stackrel{?}= \dfrac{\sin \theta}{\cos \theta}\left(\dfrac{1}{{\color{#0000FF}{\frac{1}{\cos \theta}}}}\right)
\sin \theta \stackrel{?}= \dfrac{\sin \theta}{\cos \theta}\left(\dfrac{\cos \theta}{1}\right)
\sin \theta \stackrel{?}= \dfrac{\sin \theta \cdot \cos \theta}{\cos \theta}
\sin \theta \stackrel{?}= \dfrac{\sin \theta \cdot \cancel{{\color{#FF0000}{\cos \theta}}}}{\cancel{{\color{#FF0000}{\cos \theta}}}}
sin θ = sin θ ✓
Having proved the identity sin θ = tan θsec θ, we can prove the identity csc θ= sec θtan θ.
\csc \theta \stackrel{?}= \dfrac{1}{\sin \theta}
\csc \theta \stackrel{?}= \dfrac{1}{{\color{#0000FF}{\frac{\tan \theta}{\sec \theta}}}}
csc θ = sec θ/tan θ ✓
Moreover, we will show that cot θ = 1tan θ.
\cot \theta \stackrel{?}= \dfrac{1}{\tan \theta}
\cot \theta \stackrel{?}= \dfrac{1}{{\color{#0000FF}{ \frac{\sin \theta}{\cos \theta} }}}
\cot \theta \stackrel{?}= \dfrac{\cos \theta}{\sin \theta}
cot θ = cot θ ✓
Now we are ready to move to simplifying the above fractions, which requires different methods for each case. Let's start with sin θ.
sin θ = - sqrt(65)4/94
sin θ = - sqrt(65)/9
To simplify cot θ and csc θ, we need to rationalize the denominators. Let's look at how this was done for csc θ first.
csc θ=94/- sqrt(65)4
csc θ=9/- sqrt(65)
csc θ=- 9/sqrt(65)
csc θ=- 9 sqrt(65)/sqrt(65)* sqrt(65)
csc θ=- 9 sqrt(65)/(sqrt(65))^2
csc θ=- 9 sqrt(65)/65
Let's now follow a similar procedure to rationalize the denominator of cot θ.
cot θ = 1/- sqrt(65)4
cot θ = 4/- sqrt(65)4 * 4
cot θ = 4/- sqrt(65)
cot θ = - 4/sqrt(65)
cot θ = - 4sqrt(65)/sqrt(65) * sqrt(65)
cot θ = - 4 sqrt(65)/(sqrt(65))^2
cot θ = - 4 sqrt(65)/65
