Possible To Do More Complex Trignometry Problems By Hand? Are there straightforward ways to do the following two problems by hand instead of using a calculator?  Any shortcuts (i.e. the denominator of question 5?  

Or is the expectation for these types of problems to just do them on the calculator ?  

$\cos \dfrac{23\pi}{4} 
   = \cos \left( 6\pi - \dfrac{\pi}{4} \right)
   = \cos \left(- \dfrac{\pi}{4} \right)
   = \dfrac{1}{\sqrt 2}$
$\sin \dfrac{15\pi}{4} 
   = \sin \left( 4\pi - \dfrac{\pi}{4} \right)
   = \sin\left(- \dfrac{\pi}{4} \right)
   = -\dfrac{1}{\sqrt 2}$
$$\cos \dfrac{23\pi}{4} - \sin \dfrac{15\pi}{4} = \sqrt 2$$
$$\dfrac{\cos 35^\circ}{\sin 20^\circ \cos 35^\circ + \cos 20^\circ \sin 35^\circ}
  =\dfrac{\cos 35^\circ}{\sin 55^\circ}  
  = \dfrac{\cos 35^\circ}{\cos (90^\circ - 55^\circ)}
  = 1$$
