CBSE Class 12 Math 2020 Delhi Set 2 Solved Paper | Question 9

CBSE Class 12 Math 2020 Delhi Set 2 Solved Paper

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Question : 9

Total: 11

Prove that tan[2tan−1‌

1

2

−cot−13]=‌

9

13

Solution:

Let m=tan−1‌

1

2

.
⇒‌‌tan‌m=‌

1

2

‌‌‌⋅⋅⋅⋅⋅⋅⋅(i)
and n=cot−13
⇒‌‌cot‌n=3.
⇒‌‌tan‌n=‌

1

3

‌‌‌⋅⋅⋅⋅⋅⋅⋅(ii)
∴tan[2tan−1‌

1

2

−cot−13]=tan(2m−n)

=‌

tan‌2‌m−tan‌n

1+tan‌2‌m‌tan‌n.

[

∵tan(a−b)

=‌

tan‌a−tan‌b

1+tan‌a‌tan‌b

]
=‌

‌

2‌tan‌m

1−tan2m

−tan‌n

1+‌

2‌tan‌m

1−tan2m

‌tan‌n

=‌

‌

2×

1

2

1−(

1

2

)2

−

1

3

1+‌

2×

1

2

1−(‌

1

2

)2

×‌

1

3

=‌

1

1−‌

1

4

−‌

1

3

‌=‌

‌

4

3

−

1

3

1+‌

1

1−‌

1

4

×‌

1

3

×‌

1

3

‌=‌

1

1+‌

4

9

‌=‌

9

9+4

‌=‌

9

13

.

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