Prove the following: \tan^{-1}1/3 + \tan^{-1}1/5 + \tan^{-1}1/7 + \tan^{-1}1/8 = \pi/8

CBSE Class 12 Math 2008 Solved Paper - Question 12

Question : 12 of 29

Marks: +1, -0

Prove the following:

tan^{-1}1/3 + tan^{-1}1/5

tan^{-1}1/7 + tan^{-1}1/8

π/8

Solution:

L.H.S. =

tan^{-1}1/3 + tan^{-1}1/5

tan^{-1}1/7 + tan^{-1}1/8

tan^{-1}|{1/3+1/5}/{1-(1/3)(1/5)}|

tan^{-1}|{1/7+1/8}/{1-(1/7)(1/8)}|

tan^{-1}|{{5+3}/{15}}/{{15-1}/{15}}|

tan^{-1}|{{8+7}/{56}}/{{56-1}/{56}}|

tan^{-1}|{8/{15}}/{{14}/{15}}|

tan^{-1} |{{15}/{56}}/{{55}/{56}}|

tan^{-1}(8/{14})

tan^{-1}({15}/{55})

tan^{-1}(4/7)+tan^{-1}(3/{11})

tan^{-1}({4/7+3/{11}}/{1-(4/7)(3/{11})})

tan^{-1}({{44+21}/{77}}/{{77-12}/{77}})

tan^{-1}({65}/{65})

tan^{-1}

1
=

tan^{-1} (tan π/4)

π/4

= R.H.S.
Hence proved.

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