Basic trigo - II

Geometry Level 3

If $a\sec { \theta -c\tan { \theta =d } }$ and $b\sec { \theta +d\tan { \theta =c } }$ , then which of the following relations are correct?

This question is a part of the set: Basic Trigo

ab=cd

{ a }^{ 2 }+{ b }^{ 2 }={ c }^{ 2 }+{ d }^{ 2 }

{ a }^{ 2 }+{ c }^{ 2 }={ b }^{ 2 }+{ d }^{ 2 }

{ a }^{ 2 }+{ d }^{ 2 }={ c }^{ 2 }+{ b }^{ 2 }

1 solution

Krishna Ramesh
May 18, 2014

Dividing both the equations by $\sec { \theta }$ , we get

$a-c\sin { \theta =d\cos { \theta } }$ and $b+d\sin { \theta =c\cos { \theta } }$

$\Rightarrow \quad a=d\cos { \theta +c\sin { \theta } } \\$ $and\quad b=c\cos { \theta -d\sin { \theta } } \\$

squaring and adding both equations, we get

${ a }^{ 2 }+{ b }^{ 2 }={ d }^{ 2 }\cos ^{ 2 }{ \theta } +{ c }^{ 2 }\sin ^{ 2 }{ \theta -2cd\sin { \theta } \cos { \theta \quad +\quad } } { c }^{ 2 }\cos ^{ 2 }{ \theta } +{ d }^{ 2 }\sin ^{ 2 }{ \theta +2cd\sin { \theta } \cos { \theta \quad \quad } }$

$\Rightarrow { a }^{ 2 }+{ b }^{ 2 }=\quad { c }^{ 2 }\left( \sin ^{ 2 }{ \theta } +\cos ^{ 2 }{ \theta } \right) +{ d }^{ 2 }\left( \sin ^{ 2 }{ \theta } +\cos ^{ 2 }{ \theta } \right)$

$\Rightarrow { a }^{ 2 }+{ b }^{ 2 }=\quad { c }^{ 2 }+{ d }^{ 2 }$

Basic trigo - II

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