Please solve by elimination method.

Question

Given that $a,b,c,d ⊆ R$, if $$a \sec(200°) - c \tan(200°) = d$$ and $$b \sec(200°) + d \tan(200°) = c$$ then find the value of $$\dfrac {(a^2+b^2+c^2+d^2)(\sin 20°)}{(bd-ac)}.$$ I got the relation of c^2+d^2.

Answer 1

Let $$\sec(200^{\circ})=A$$ and $$\tan(200^{\circ})=B$$ then our system is given by $$aA-cB=d$$ $$Ab+Bd=c$$ From the second equation we get $$aA-(Ab+BdB)B=d$$ solving for $d$ we obtain $$d=\frac{aA-ABb}{B^2+1}=\frac{A(a-Bb)}{B^2+1}$$ using this equation we get $c$: $$c=\frac{A(Ba-b)}{B^2+1}$$ Can you finish?

Answer 2

Let $200^\circ=t$

multiply out both relations by $\cos t$ to find

$$b\sin t+d\cos t=a$$

$$-d\sin t+c\cos t=b$$

Solve for $\sin t$