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Question

Question: \(\sin \left( 4 \tan ^ { - 1 } \frac { 1 } { 3 } \right) =\)...

sin⁡(4tan⁡−113)=\sin \left( 4 \tan ^ { - 1 } \frac { 1 } { 3 } \right) =

A

1225\frac { 12 } { 25 }

B

2425\frac { 24 } { 25 }

C

15\frac { 1 } { 5 }

D

None of these

Answer

2425\frac { 24 } { 25 }

Explanation

Solution

sin⁡(4tan⁡−113)=sin⁡[2tan⁡−1(2/31−(1/9))]\sin \left( 4 \tan ^ { - 1 } \frac { 1 } { 3 } \right) = \sin \left[ 2 \tan ^ { - 1 } \left( \frac { 2 / 3 } { 1 - ( 1 / 9 ) } \right) \right]

=sin⁡[2tan⁡−134]=sin⁡sin⁡−1(2×(3/4)1+(9/16))= \sin \left[ 2 \tan ^ { - 1 } \frac { 3 } { 4 } \right] = \sin \sin ^ { - 1 } \left( \frac { 2 \times ( 3 / 4 ) } { 1 + ( 9 / 16 ) } \right)

=32×1625=2425= \frac { 3 } { 2 } \times \frac { 16 } { 25 } = \frac { 24 } { 25 } (∵2tan⁡−1x=sin⁡−12x1+x2)\left( \because 2 \tan ^ { - 1 } x = \sin ^ { - 1 } \frac { 2 x } { 1 + x ^ { 2 } } \right)