Let us consider LHS: \[si{{n}^{2}}~\pi /18\text{ }+\text{ }si{{n}^{2}}~\pi /9\] \[+\text{ }si{{n}^{2}}~7\pi /18\text{ }+\text{ }si{{n}^{2}}~4\pi /9\] Or, \[si{{n}^{2}}~\pi /18\text{ }+\text{...

Solution: As per the given question, \[1\text{ }=\text{ }RHS\] \[\therefore LHS\text{ }=\text{ }RHS\] Hence proved.

(i) (ii) Solution: (i) \[1\text{ }=\text{ }RHS\] \[\therefore LHS\text{ }=\text{ }RHS\] Hence proved. (ii) \[\left\{ 1+cotx\left-( -cosecx \right) \right\}\left\{ 1+cotx+\left( -cosecx \right)...

(i) (ii)   Solution: (i) \[1\text{ }=\text{ }RHS\] \[\therefore LHS\text{ }=\text{ }RHS\] Hence proved. (ii) \[1\text{ }+\text{ }1\] \[2\text{ }=\text{ }RHS\] \[\therefore LHS\text{ }=\text{...

(i) \[3\text{ }sin\text{ }\pi /6\text{ }sec\text{ }\pi /3\text{ }-\text{ }4\text{ }sin\text{ }5\pi /6\text{ }cot\text{ }\pi /4\text{ }=\text{ }1\] Let us consider LHS: \[3\text{ }sin\text{ }\pi...

(i) \[cos\text{ }{{570}^{o}}~sin\text{ }{{510}^{o}}~+\text{ }sin\text{ }(-{{330}^{o}})\text{ }cos\text{ }(-{{390}^{o}})\] \[=\text{ }0\] Let us consider LHS: \[cos\text{ }{{570}^{o}}~sin\text{...

(i) \[cos\text{ }{{24}^{o}}~+\text{ }cos\text{ }{{55}^{o}}~+\text{ }cos\text{ }{{125}^{o}}~+\text{ }cos\text{ }{{204}^{o}}~\] \[~+\text{ }cos\text{ }{{300}^{o}}~=\text{ }1/2\] Let us consider LHS:...

(i) \[tan\text{ }{{225}^{o}}~cot\text{ }{{405}^{o}}~+\text{ }tan\text{ }{{765}^{o}}~cot\text{ }{{675}^{o}}~=\text{ }0\] Let us consider LHS: \[tan\text{ }225{}^\circ ~\text{ }cot\text{ }405{}^\circ...

(i) \[cos\text{ }39\pi /4\] \[cos\text{ }39\pi /4\text{ }=\text{ }cos\text{ }{{1755}^{o}}\] \[=\text{ }cos\text{ }{{\left( 90\times 19\text{ }+\text{ }45 \right)}^{o}}\] Since,\[{{1755}^{o}}\] lies...

(i) \[cos\text{ }19\pi /4\] \[cos\text{ }19\pi /4\text{ }=\text{ }cos\text{ }{{855}^{o}}\] \[=\text{ }cos\text{ }{{\left( 90\times 9\text{ }+\text{ }45 \right)}^{o}}\] Since,\[{{855}^{o}}\] lies in...

(i) \[cosec\text{ }\left( -20\pi /3 \right)\] \[cosec\text{ }\left( -20\pi /3 \right)\] \[=\text{ }cosec\text{ }{{\left( -1200 \right)}^{o}}\] Or, \[=\text{ }-\text{ }cosec\text{ }{{\left( 1200...

(i) \[cos\text{ }19\pi /6\] \[cos\text{ }19\pi /6\text{ }=\text{ }cos\text{ }{{570}^{o}}\] \[=\text{ }cos\text{ }{{\left( 90\times 6\text{ }+\text{ }30 \right)}^{o}}\] Since,\[{{570}^{o}}\] lies in...

(i) \[tan\text{ }7\pi /4\]  \[tan\text{ }7\pi /4\text{ }=\text{ }tan\text{ }{{315}^{o}}\] \[=\text{ }tan\text{ }{{\left( 90\times 3\text{ }+\text{ }45 \right)}^{o}}\] Since,\[{{315}^{o}}\] lies in...

(i) \[tan\text{ }11\pi /6\] \[tan\text{ }11\pi /6\text{ }=\text{ }{{\left( 11/6\text{ }\times \text{ }180 \right)}^{o}}\] \[=\text{ }{{330}^{o}}\] Since,\[{{330}^{o}}\] lies in the \[IV\text{...

(i) \[sin\text{ }5\pi /3\] \[5\pi /3\text{ }=\text{ }{{\left( 5\pi /3\text{ }\times \text{ }180 \right)}^{o}}\] \[=\text{ }{{300}^{o}}\] Or, \[=\text{ }{{\left( 90\times 3\text{ }+\text{ }30...

Solution: According to the given question: \[cos\text{ }x=\text{ }-3/5\text{ }and\text{ }\pi \text{ }<x\text{ }<\text{ }3\pi /2\] We know that in the third quadrant, \[tan\text{ }x\text{...

According to the given question: \[Sin\text{ }x\text{ }+\text{ }cos\text{ }x\text{ }=\text{ }0\text{ }and\text{ }x\]lies in fourth quadrant. \[Sin\text{ }x~=\text{ }-cos\text{ }x\] \[Sin\text{...

According to the given question: \[sin\text{ }x\text{ }=\text{ }3/5,\text{ }tan\text{ }y\text{ }=\text{ }1/2\] And \[\pi /2\text{ }<\text{ }x<\text{ }\pi <\text{ }y<\text{ }3\pi /2~\] We...

According to the given question: \[Sin\text{ }x\text{ }=\text{ }12/13\text{ }and\text{ }x\]lies in the second quadrant. We know, in second quadrant, \[sin\text{ }x\text{ }and\text{ }cosec\text{...

(i) \[tan\text{ }x\text{ }=\text{ }3/4,\text{ }x\text{ }in\text{ }quadrant\text{ }III\] In third quadrant, \[tan\text{ }x\text{ }and\text{ }cot\text{ }x\] are positive. \[sin\text{ }x,\text{...

(i) \[cot\text{ }x\text{ }=\text{ }12/5,\text{ }x\text{ }in\text{ }quadrant\text{ }III\] In third quadrant, \[tan\text{ }x\text{ }and\text{ }cot\text{ }x\]are positive. \[sin\text{ }x,\text{...