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Question

Prove that:

(i)tan225cot405+tan765cot675=0

(ii)sin8π3cos23π6+cos13π3sin35π6=12

(iii)cos24+cos55+cos125+cos204+cos300=12

(iv)tan(225)cot(405)tan(765)cot(675)=0

(v)cos570sin510+sin(330)cos(390)=0

(vi)tan11π32sin4π634cosec2π4+4cos217π6=3432

(vii)3sinπ6secπ34sin5π5cotπ4=1


Solution

(i)LHS=tan225cot405+tan765cot675

=tan(π+π4)cot(2π+π4)+tan(4π+π4)cot(4ππ4)

=tanπ4.cotπ4+tanπ4×(cotπ4)

[cot(4ππ4)=cotπ4]

=1.1 +1.(-1)

=RHS Hence proved.

(ii)LHS=sin8π3cos23π6+cos13π3sin35π6

=sin(3ππ3)cos(4ππ6)+cos(4π+π3)sin(6ππ6)

=sinπ3cosπ6+cosπ3(sinπ6)

[sin(6πθ)=sinθ]

=32×322+12×(12)

=3414

=24

=12=RHS Hence proved.

(iii)LHS=cos24+cos55+cos125+cos204+cos300

=cos24+cos204+cos55+cos125+cos300

=cos24+cos(π+24)+cos55+cos(π55)+cos(2ππ3)

=cos24cos24+cos55cos55+cosπ3

=cosπ3

=12=RHS Hence proved.

(iv)LHS=tan(225)cot(405)tan(765)cot(675)

=tan225(cot405)+tan765cot765

[tan(θ)=tanθ and cot(θ)=cotθ]

=tan(π+π4)cot(2ππ4)+tan(4π+π4)cot(4ππ4)

=tanπ4cotπ4+tanπ4×(cotπ4)[cot(4πθ)=cotθ]

=1.1+1(-1)=1-1=0=RHS Hence proved.

(v)LHS=cos 570 sin 510+sin(330)cos(390)

=cos(3π+π6)sin(3ππ6)sin330cos390

[sin(θ)=sinθandcos(θ)=cosθ]

=cosπ6sinπ6sin(2ππ6)cos(2π+π6)

=sinπ6cosπ6+sinπ6cosπ6[sin(2πθ)=sinθ]

0=RHS Hence proved.

(vi)LHS=tan11π32sin4π634cosec2π4+4cos217π6

=tan(4ππ3)2sin2π334×(2)2+4cos2(3ππ6)

=tanπ32sin(ππ3)34×2+4cos2π6

(tan(4ππ3)=tanπ3,cos(3ππ6)=cosπ6)

=32sinπ332+4×(32)2

=32×3232+4×34

=3332+3=233+62=2332

=3432 =RHS Hence proved.

(vii)LHS=3sinπ6secπ34sin5π6cotπ4

=3×12×24sin(ππ6)×1

=34siinπ6[sin(πθ)=sinθ]

=34×12=32=1 =RHS Hence proved.


Mathematics
RD Sharma
Standard XI

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