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Question

Prove:
2sin135tan11731=π4

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Solution

2sin135tan11731=π4

Let sin135=a

sina=35

sin2a=925

cos2a=1sin2a=1925=25925=1625

cosa=45

tana=sinacosa=3545=34

a=tan134

sin135=tan134

Hence
2 sin135tan11731=2 tan134tan11731

We know that,
2 tan1x=tan1(2x1x2)

L.H.S=tan1⎜ ⎜ ⎜ ⎜ ⎜2×341(34)2⎟ ⎟ ⎟ ⎟ ⎟tan11731

L.H.S=tan1⎜ ⎜ ⎜641916⎟ ⎟ ⎟tan11731

L.H.S=tan1⎜ ⎜ ⎜6416916⎟ ⎟ ⎟tan11731

L.H.S=tan1247tan11731

We know that tan1xtan1y=tan1(xy1+xy)

L.H.S=tan1⎜ ⎜ ⎜24717311+247×1731⎟ ⎟ ⎟

L.H.S=tan1⎜ ⎜ ⎜7441192171+408217⎟ ⎟ ⎟

L.H.S=tan1⎜ ⎜ ⎜744119217217+408217⎟ ⎟ ⎟

L.H.S=tan1⎜ ⎜ ⎜625217625217⎟ ⎟ ⎟

L.H.S=tan11=π4=R.H.S
Hence proved.

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