Ratios of complementary angles

How to use it

1. 1Scan a long expression for angles that add to $90^{\circ }$; each such pair can be brought to a common angle, after which the terms cancel or collapse.
2. 2A ratio over the co-ratio of its complement is always $1$, which is what makes $\frac{\cos 75^{\circ }}{\sin 15^{\circ }}$ and $\sin 59^{\circ }\sec 31^{\circ }$ equal to $1$ with no table at all.
3. 3Convert every term to a single angle before reaching for a Pythagorean identity.

Step-by-step

1. 1
   $\cos 50^{\circ }=\sin \left(90^{\circ }-50^{\circ }\right)=\sin 40^{\circ }$

   Why: $40^{\circ }$ and $50^{\circ }$ are complementary.

2. 2
   $\frac{\sin 40^{\circ }}{\cos 50^{\circ }}=1$
3. 3
   $\tan 70^{\circ }=\cot 20^{\circ }\Rightarrow \tan 20^{\circ }\tan 70^{\circ }=\tan 20^{\circ }\cot 20^{\circ }=1$

   Why: $20^{\circ }$ and $70^{\circ }$ are complementary, and a tangent times its own cotangent is $1$.

4. 4
   $1+1=2$