Prove that: (i)tan225∘cot405∘+tan765∘cot675∘=0 (ii)sin8π3cos2

1.	Prove that: (i)tan225∘cot405∘+tan765∘cot675∘=0 (ii)sin8π3cos23π6+cos13π3sin35π6=12 (iii)cos24∘+cos55∘+cos125∘+cos204∘+cos300∘=12 (iv)tan(−225∘)cot(−405∘)−tan(−765∘)cot(675∘)=0 (v)cos570∘sin510∘+sin(−330∘)cos(−390∘)=0 (vi)tan11π3−2sin4π6−34cosec2π4+4cos217π6=3−4√32 (vii)3sinπ6secπ3−4sin5π5cotπ4=1
Answer» Prove that: $(i) t a n 225^{\circ} c o t 405^{\circ} + t a n 765^{\circ} c o t 675^{\circ} = 0$ $(i i) s i n \frac{8 π}{3} c o s \frac{23 π}{6} + c o s \frac{13 π}{3} s i n \frac{35 π}{6} = \frac{1}{2}$ $(i i i) c o s 24^{\circ} + c o s 55^{\circ} + c o s 125^{\circ} + c o s 204^{\circ} + c o s 300^{\circ} = \frac{1}{2}$ $(i v) t a n (- 225^{\circ}) c o t (- 405^{\circ}) - t a n (- 765^{\circ}) c o t (675^{\circ}) = 0$ $(v) c o s 570^{\circ} s i n 510^{\circ} + s i n (- 330^{\circ}) c o s (- 390^{\circ}) = 0$ $(v i) t a n \frac{11 π}{3} - 2 s i n \frac{4 π}{6} - \frac{3}{4} c o s e c^{2} \frac{π}{4} + 4 c o s^{2} \frac{17 π}{6} = \frac{3 - 4 \sqrt{3}}{2}$ $(v i i) 3 s i n \frac{π}{6} s e c \frac{π}{3} - 4 s i n \frac{5 π}{5} c o t \frac{π}{4} = 1$

Prove that: (i)tan225∘cot405∘+tan765∘cot675∘=0 (ii)sin8π3cos23π6+cos13π3sin35π6=12 (iii)cos24∘+cos55∘+cos125∘+cos204∘+cos300∘=12 (iv)tan(−225∘)cot(−405∘)−tan(−765∘)cot(675∘)=0 (v)cos570∘sin510∘+sin(−330∘)cos(−390∘)=0 (vi)tan11π3−2sin4π6−34cosec2π4+4cos217π6=3−4√32 (vii)3sinπ6secπ3−4sin5π5cotπ4=1

Answer»

Prove that:

$(i) t a n 225^{\circ} c o t 405^{\circ} + t a n 765^{\circ} c o t 675^{\circ} = 0$

$(i i) s i n \frac{8 π}{3} c o s \frac{23 π}{6} + c o s \frac{13 π}{3} s i n \frac{35 π}{6} = \frac{1}{2}$

$(i i i) c o s 24^{\circ} + c o s 55^{\circ} + c o s 125^{\circ} + c o s 204^{\circ} + c o s 300^{\circ} = \frac{1}{2}$

$(i v) t a n (- 225^{\circ}) c o t (- 405^{\circ}) - t a n (- 765^{\circ}) c o t (675^{\circ}) = 0$

$(v) c o s 570^{\circ} s i n 510^{\circ} + s i n (- 330^{\circ}) c o s (- 390^{\circ}) = 0$

$(v i) t a n \frac{11 π}{3} - 2 s i n \frac{4 π}{6} - \frac{3}{4} c o s e c^{2} \frac{π}{4} + 4 c o s^{2} \frac{17 π}{6} = \frac{3 - 4 \sqrt{3}}{2}$

$(v i i) 3 s i n \frac{π}{6} s e c \frac{π}{3} - 4 s i n \frac{5 π}{5} c o t \frac{π}{4} = 1$