Vanuatu
Simplify ^@ \dfrac{ \cos^2 10 ^\circ + \cos^2 80 ^\circ } { \sin^2 10 ^\circ + \sin^2 80 ^\circ } + \dfrac{ \sin (90 ^\circ - \theta ) \sin \theta } { \tan \theta } + \dfrac{ \cos (90 ^\circ - \theta ) \cos \theta } { \cot \theta } ^@
Answer:
^@ 2 ^@
- ^@ \begin{align} & \dfrac{ \cos^2 10 ^\circ + \cos^2 80 ^\circ } { \sin^2 10 ^\circ + \sin^2 80 ^\circ } + \dfrac{ \sin (90 ^\circ - \theta ) \sin \theta } { \tan \theta } + \dfrac{ \cos (90 ^\circ - \theta ) \cos \theta } { \cot \theta } \\ = & \dfrac{ cos^210^\circ + cos^2(90^\circ - 10^\circ) } { sin^210^\circ + sin^2(90^\circ - 10^\circ) } + \dfrac{ cos\theta sin\theta } { tan\theta } + \dfrac{ sin\theta cos\theta } { cot\theta } && \begin{cases} sin(90^\circ - \theta) = cos \theta \text{ and } \\ cos( 90^\circ - \theta) = sin \theta \end{cases} \\ = & \dfrac{ cos^210^\circ + sin^10^\circ } { sin^210^\circ + cos^210^\circ } + \dfrac{ cos\theta sin\theta } { tan\theta } + \dfrac{ sin\theta cos\theta } { cot\theta } \\ = & 1 + cos^2\theta + sin^2\theta \\ = & 1 + 1 \\ = & 2 \end{align} ^@
