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Question

For 0<θ<π2, the solution (s) of
6m=1cosec(θ+(m1)π4)cosec(θ+mπ4)=42 is/are

A
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B
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C
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D
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Solution

The correct options are
C
D
For 0<θ<π2
6m=1cosec(θ+(m1)π4)cosec(θ+mπ4)=42
6m=11sin(θ+(m1)π4)sin(θ+mπ4)=42
6m=1sin[θ+mπ4(θ+(m1)π4)]sin(θ+(m1)π4)sin(θ+mπ4)=4
6m=1[cot(θ+(m1)π4)cot(θ+mπ4)]=4
cot(θ)cot(θ+π4)+cot(θ+π4)
cot(θ+2π4)+...+cot(θ+5π4)cot(θ+6π4)=4
cotθcot(3π2+θ)=4
cotθ+tanθ=4
tan2θ4tanθ+1=0
(tanθ2)23=0
(tanθ2+3)(tanθ23)=0
tanθ=23ortanθ=2+3
θ=π12;θ=5π12
θ(0,θ2)

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