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Question

Prove that:

(i)tan720cos270sin150cos120=14

(ii)sin780sin480+cos120sin150=12

(iii)sin780sin120+cos240sin390=12

(iv)sin600cos390+cos480sin150=1

(v)tan250cot405+tan765cot675=0


Solution

(i)LHS=tan720cos270sin150cos120

=tan4πcos(3π2)sin(ππ6)

cos(π2+π6)[π=180]

=00sinπ6(sinπ6)

[tannπ=0forallnZandcos3π2=0]

=sin2π6

=(12)2=14=RHS

Hence proved.

(ii)LHS=sin780sin480+cos120sin150

=sin(4π+π3)sin(3ππ3)+cos

(π2+π6)sin(ππ6)

[π=180]

=sinπ3×sinπ3+(sinπ6)sinπ6

sin(4π+π3)=sinπ3and sin(3π+π3)=sinπ3

=32×3212×12=3414

=24=12=RHS Hence proved.

(iii)LHS=sin780sin120+cos240sin390

=sin(4π+π3)sin(π2+π6)+cos(π+π6)sin(2π+π6)

=sinπ3×cosπ6cosπ3×(+sinπ6)

=32×3212×12

=3414

=12=12=RHS Hence proved.

(iv)LHS=sin600cos390+cos480sin150

=sin(3π+π3)cos(2π+π6)+cos(3ππ3)sin(ππ6)

=sinπ3cosπ6cosπ3sinπ6

sin(3π+π3)=sinπ3 and cos(3ππ3)=cosπ3

=32×sqrt3212×12=3414=44

=-1 =RHS Hence proved.=

(v)LHS=tan225cot405+tan765cot675

=tan(π+π4)cot(2π+π4)(4π+π4)cot(4ππ4)

=tanπ4cotπ4+tanπ4(cotπ4)

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


Mathematics
RD Sharma
Standard XI

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