Relations and Functions

Relations and Functions

Question

If a, b, c are in A.P. show that cot straight A over 2 comma space cot straight B over 2 space and space cot straight C over 2 are also in A.P.

Answer

If a, b, c are in A.P.,
then                              2b = a + c                                          ...(i)
Now,   space space space cot open parentheses straight A over 2 close parentheses comma space space cot open parentheses straight B over 2 close parentheses comma space space cot open parentheses straight C over 2 close parentheses will be in A.P.
if                      2 space cot space open parentheses straight B over 2 close parentheses space equals space cot space open parentheses straight A over 2 close parentheses plus cot space open parentheses straight C over 2 close parentheses
if              2 square root of fraction numerator straight s left parenthesis straight s minus straight b right parenthesis over denominator left parenthesis straight s minus straight c right parenthesis space left parenthesis straight s minus straight a right parenthesis end fraction end root space equals space square root of fraction numerator straight s left parenthesis straight s minus straight a right parenthesis over denominator left parenthesis straight s minus straight b right parenthesis space left parenthesis straight s minus straight c right parenthesis end fraction end root space plus space square root of fraction numerator straight s left parenthesis straight s minus straight c right parenthesis over denominator left parenthesis straight s minus straight a right parenthesis space left parenthesis straight s minus straight b right parenthesis end fraction end root
if          fraction numerator 2 square root of straight s left parenthesis straight s minus straight b right parenthesis end root cross times square root of straight s left parenthesis straight s minus straight b right parenthesis end root over denominator square root of straight s left parenthesis straight s minus straight a right parenthesis space left parenthesis straight s minus straight b right parenthesis space left parenthesis straight s minus straight c right parenthesis end root end fraction space equals space fraction numerator square root of straight s left parenthesis straight s minus straight a right parenthesis end root space cross times space square root of straight s left parenthesis straight s minus straight a right parenthesis end root over denominator square root of straight s left parenthesis straight s minus straight a right parenthesis space left parenthesis straight s minus straight b right parenthesis space left parenthesis straight s minus straight c right parenthesis end root end fraction plus fraction numerator square root of straight s left parenthesis straight s minus straight c right parenthesis end root cross times square root of straight s left parenthesis straight s minus straight c right parenthesis end root over denominator square root of straight s left parenthesis straight s minus straight a right parenthesis space left parenthesis straight s minus straight b right parenthesis space left parenthesis straight s minus straight c right parenthesis end root end fraction
if                     2s (s - b) = s (s - a) + s (s - c)
if                     2 (s - b)  = (s - a) + (s - c)
if                      2s - 2b   = 2s - a - c
if                a + b + c - 2b  = a + b + c - a - c
if                    a + c - b = b
if                            2b = a + c
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#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
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#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>
if   a, b, c are in A.P., which is true
Hence,    cot straight A over 2 comma space space cot straight B over 2 comma space cot straight C over 2 are in A.P.

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