Principle of Mathematical Induction

Principle of Mathematical Induction

Question

If cos (A + B) sin (C - D) = cos (A - B) sin (C + D)

Show that:  tan A tan B tan C + tan D = 0

Answer

cos (A + B) sin (C - D) = cos (A - B)  sin (C + D)
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Applying componendo and dividendo, we get
space space fraction numerator cos left parenthesis straight A plus straight B right parenthesis space plus space cos left parenthesis straight A minus straight B right parenthesis over denominator cos left parenthesis straight A plus straight B right parenthesis space minus space cos left parenthesis straight A minus straight B right parenthesis end fraction space equals space fraction numerator sin left parenthesis straight C plus straight D right parenthesis space plus space sin left parenthesis straight C minus straight D right parenthesis over denominator sin left parenthesis straight C plus straight D right parenthesis space minus space sin left parenthesis straight C minus straight D right parenthesis end fraction
rightwards double arrow              fraction numerator 2 cosA space cosB over denominator negative 2 sinA space sinB end fraction space equals space fraction numerator 2 space sinC space cosD over denominator 2 space cosC space sinD end fraction           rightwards double arrow  <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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rightwards double arrow                        fraction numerator negative 1 over denominator cot space straight D end fraction space equals space fraction numerator tan space straight C over denominator cot space straight A space cot space straight B end fraction             rightwards double arrow   negative space tan space straight D space equals space tanA space tanB space tanC
Hence,     tanA space space tanB space space tanC space plus space tanD space equals space 0

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