Brazil
Simplify: ^@ \dfrac{ \cos^2 20 ^\circ + \cos^2 70 ^\circ } { \sin^2 20 ^\circ + \sin^2 70 ^\circ } + \cos (20 ^\circ + \theta ) - \sin (70 ^\circ - \theta ) ^@
Answer:
^@ 1 ^@
- ^@ \begin{align} & \dfrac{ \cos^2 20 ^\circ + \cos^2 70 ^\circ } { \sin^2 20 ^\circ + \sin^2 70 ^\circ } + \cos (20 ^\circ + \theta ) - \sin (70 ^\circ - \theta ) \\ = & \dfrac{ cos^220^\circ + sin^220^\circ } { sin^220^\circ + cos^220^\circ} + cos(20^\circ + \theta) - sin\left(90^\circ - (20^\circ + \theta)\right) && \begin{cases} sin(90^\circ - \theta) = cos \theta \text{ and } \\ cos( 90^\circ - \theta) = sin \theta \end{cases} \\ = & \dfrac{ cos^220^\circ + sin^220^\circ } { sin^220^\circ + cos^220^\circ} + cos(20^\circ + \theta) - cos(20^\circ + \theta) \\ = & 1 + 0 \\ = & 1 \end{align} ^@
