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Question

If θ=30, verify that :
(i) tan2θ=2tanθ1tan2θ

(ii) tan2θ=4tanθ1+tan2θ

(iii) cos2θ=1tan2θ1+tan2θ

(iv) cos3θ=4cos3θ3cosθ

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Solution

Given θ=30
To verify,
(i) tan2θ=2tanθ1tan2θ

tan2(30)=2tan301tan230

tan(60)=2tan301tan230

3=2×131(13)2

3=231(13)

3=2323

3=3



(ii) tan2θ=4tanθ1+tan2θ

We have, 1+tan2θ=sec2θ

tan2θ=4tanθsec2θ

tan2(30)=4tan30sec230

tan(60)=4tan30sec230

3=413(23)2

3=4343

3=33

3=3



(iii) cos2θ=1tan2θ1+tan2θ

cos2(30)=1tan2301+tan230

cos60=1tan2301+tan230

12=1(13)21+(13)2

12=313+1

12=24

12=12



(iv) cos3θ=4cos3θ3cosθ

cos3(30)=4cos3303cos30

cos(90)=4(cos330)3cos30

0=4(32)33(32)

0=4(338)3(32)

0=3(32)3(32)

0=0


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