A line OA of length r starts from its initial position OX and traces an angle $$AOB=$$α in the anticlockwise direction. It then traces back in the clockwise direction at an angle BOC $$-$$ $$30$$ (where$$ α$$3$$θ$$). Z is the foot of the perpendicular form C on OA. Also, $$sin3$$⁡θ$$CL=cos3$$⁡θ$$OL=1$$. $$1$$−rcos⁡αrsin⁡α is equal to $$2rsin$$⁡α$$1+2rcos$$⁡α is equal $$to2r2$$−$$1r$$ is equal to

Question

A line OA of length r starts from its initial position OX and traces an angle $$AOB=$$α in the anticlockwise direction. It then traces back in the clockwise direction at an angle BOC $$-$$ $$30$$ (where$$ α>$$3$$θ$$). Z is the foot of the perpendicular form C on OA. Also, $$sin3$$⁡θ$$CL=cos3$$⁡θ$$OL=1$$. $$1$$−rcos⁡αrsin⁡α is equal to $$2rsin$$⁡α$$1+2rcos$$⁡α is equal $$to2r2$$−$$1r$$ is equal to

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$$OL=rcos$$⁡$$($$α−$$3$$θ$$)CL=rsin$$⁡$$($$α−$$3$$θ$$)So$$, $$sin3$$⁡θrsin⁡$$($$α−$$3$$θ$$)=cos3$$⁡θrcos⁡$$($$α−$$3$$θ$$)=1$$⇒$$sin3$$⁡θ$$sin$$⁡$$($$α−$$3$$θ$$)=cos3$$⁡θ$$cos$$⁡$$($$α−$$3$$θ$$)=rNow$$ $$cos4$$⁡θ−$$sin4$$⁡θ $$=$$$$cos$$⁡θ(rcos$$⁡$$($$α−$$3$$θ$$))−$$sin$$⁡θ(rsin$$⁡$$($$α−$$3$$θ$$))⇒$$cos$$⁡$$2$$θ$$=rcos$$⁡$$($$α−$$2$$θ$$)$$⇒$$cos$$⁡$$2$$θ$$=r($$$$cos$$⁡α$$cos$$⁡$$2$$θ$$+$$$$sin$$⁡α$$sin$$⁡$$2$$θ$$)$$⇒(1$$−rcos⁡α$$)$$cos$$⁡$$2$$θ$$=rsin$$⁡α$$sin$$⁡$$2$$θ⇒$$1$$−rcos⁡αrsin⁡α$$=$$$$tan$$⁡$$2$$θ.....(i)cos3$$⁡θ$$sin$$⁡θ$$+sin3$$⁡θ$$cos$$⁡θ⇒$$sin$$⁡θ$$cos$$⁡θ$$=rsin$$⁡$$($$α−$$2$$θ$$)⇒$$sin$$⁡$$2$$θ$$=2r$$[$$sin$$⁡α$$cos$$⁡$$2$$θ−$$cos$$⁡α$$sin$$⁡$$2$$θ]⇒(1+2rcos$$⁡α$$)$$sin$$⁡$$2$$θ$$=2rsin$$⁡α$$cos$$⁡$$2$$θ⇒$$1+2rcos$$⁡α$$2rsin$$⁡α$$=$$$$cot$$⁡$$2$$θ.....(ii)From$$ $$(1) and (2),we $$get1$$−rcos⁡αrsin⁡α$$=2rsin$$⁡α$$1+2rcos$$⁡α⇒$$1$$−rcos⁡α$$+2rcos$$⁡α−$$2r2cos2$$⁡α$$=2r2sin2$$⁡α⇒$$2r2$$−$$1r=$$$$cos$$⁡α$$cos4$$θ$$-sin4$$θ$$=$$$$cos$$⁡θ(rcos$$⁡$$($$α−$$3$$θ$$))−$$sin$$⁡θ(rsin$$⁡$$($$α−$$3$$θ$$))⇒$$cos$$⁡$$2$$θ$$=rcos$$⁡$$($$α−$$2$$θ$$)$$⇒$$cos$$⁡$$2$$θ$$=r($$$$cos$$⁡α$$cos$$⁡$$2$$θ$$+$$$$sin$$⁡α$$sin$$⁡$$2$$θ$$)$$⇒(1$$−rcos⁡α$$)$$cos$$⁡$$2$$θ$$=rsin$$⁡α$$sin$$⁡$$2$$θ⇒$$1$$−rcos⁡αrsin⁡α$$=$$$$tan$$⁡$$2$$θ.....(i)cos3$$⁡θ$$sin$$⁡θ$$+sin3$$⁡θ$$cos$$⁡θ⇒$$sin$$⁡θ$$cos$$⁡θ$$=rsin$$⁡$$($$α−$$2$$θ$$)⇒$$sin$$⁡$$2$$θ$$=2r$$[$$sin$$⁡α$$cos$$⁡$$2$$θ−$$cos$$⁡α$$sin$$⁡$$2$$θ]⇒(1+2rcos$$⁡α$$)$$sin$$⁡$$2$$θ$$=2rsin$$⁡α$$cos$$⁡$$2$$θ⇒$$1+2rcos$$⁡α$$2rsin$$⁡α$$=$$$$cot$$⁡$$2$$θ.....(ii)cos4$$θ$$-sin4$$θ$$=$$$$cos$$⁡θ$$(rcos$$⁡$$($$α−$$3$$θ$$))−$$sin$$⁡θ(rsin$$⁡$$($$α−$$3$$θ$$))⇒$$cos$$⁡$$2$$θ$$=rcos$$⁡$$($$α−$$2$$θ$$)$$⇒$$cos$$⁡$$2$$θ$$=r($$$$cos$$⁡α$$cos$$⁡$$2$$θ$$+$$$$sin$$⁡α$$sin$$⁡$$2$$θ$$)$$⇒(1$$−rcos⁡α$$)$$cos$$⁡$$2$$θ$$=rsin$$⁡α$$sin$$⁡$$2$$θ⇒$$1$$−rcos⁡αrsin⁡α$$=$$$$tan$$⁡$$2$$θ.....(i)cos3$$⁡θ$$sin$$⁡θ$$+sin3$$⁡θ$$cos$$⁡θ⇒$$sin$$⁡θ$$cos$$⁡θ$$=rsin$$⁡$$($$α−$$2$$θ$$)⇒$$sin$$⁡$$2$$θ$$=2r$$[$$sin$$⁡α$$cos$$⁡$$2$$θ−$$cos$$⁡α$$sin$$⁡$$2$$θ]⇒(1+2rcos$$⁡α$$)$$sin$$⁡$$2$$θ$$=2rsin$$⁡α$$cos$$⁡$$2$$θ⇒$$1+2rcos$$⁡α$$2rsin$$⁡α$$=$$$$cot$$⁡$$2$$θ.....(ii)from$$ $$(i) and $$(ii)1$$−rcos⁡αrsin⁡α$$=2rsin$$⁡α$$1+2rcos$$⁡α⇒$$1$$−rcos⁡α$$+2rcos$$⁡α−$$2r2cos2$$⁡α$$=2r2sin2$$⁡α⇒$$2r2$$−$$1r=$$$$cos$$⁡α

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