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Trigonometric Functions

Short Answer Type

121.

Prove the following:

$space space space cot space 4 straight x space left parenthesis sin space 5 straight x space plus space sin space 3 straight x right parenthesis space equals space cot space straight x space left parenthesis sin space 5 straight x space minus space sin space 3 straight x right parenthesis$

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122.

Prove the following:

$sin space 2 straight x space plus space 2 space sin space 4 straight x space plus space sin space 6 straight x space equals space 4 cos squared space straight x space sin space 4 straight x$

107 Views

123.

Prove that:

$2 space cos straight pi over 13 space cos fraction numerator 9 straight pi over denominator 13 end fraction plus cos fraction numerator 3 straight pi over denominator 13 end fraction plus cos fraction numerator 5 straight pi over denominator 13 end fraction space equals space 0$

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124.

Prove that:

$cosα space plus space cosβ space plus space cosγ space plus space cos space left parenthesis straight alpha space plus space straight beta space plus space straight gamma right parenthesis space equals space 4 cos open parentheses fraction numerator straight alpha plus straight beta over denominator 2 end fraction close parentheses space cos open parentheses fraction numerator straight beta plus straight gamma over denominator 2 end fraction close parentheses space cos fraction numerator straight gamma plus straight alpha over denominator 2 end fraction.$

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125.
If cos (A + B) sin (C - D) = cos (A - B) sin (C + D)

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

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

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Applying componendo and dividendo, we get

$space space fraction numerator cos left parenthesis straight A plus straight B right parenthesis space plus space cos left parenthesis straight A minus straight B right parenthesis over denominator cos left parenthesis straight A plus straight B right parenthesis space minus space cos left parenthesis straight A minus straight B right parenthesis end fraction space equals space fraction numerator sin left parenthesis straight C plus straight D right parenthesis space plus space sin left parenthesis straight C minus straight D right parenthesis over denominator sin left parenthesis straight C plus straight D right parenthesis space minus space sin left parenthesis straight C minus straight D right parenthesis end fraction$

$rightwards double arrow$ $fraction numerator 2 cosA space cosB over denominator negative 2 sinA space sinB end fraction space equals space fraction numerator 2 space sinC space cosD over denominator 2 space cosC space sinD end fraction$ $rightwards double arrow$ $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

$rightwards double arrow$ $fraction numerator negative 1 over denominator cot space straight D end fraction space equals space fraction numerator tan space straight C over denominator cot space straight A space cot space straight B end fraction$ $rightwards double arrow$ $negative space tan space straight D space equals space tanA space tanB space tanC$

Hence, $tanA space space tanB space space tanC space plus space tanD space equals space 0$

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126.

If $sin space straight theta space equals space straight n space sin space left parenthesis straight theta space plus space 2 straight alpha right parenthesis comma$ show that $tan space left parenthesis straight theta space plus space straight alpha right parenthesis space equals space fraction numerator 1 plus straight n over denominator 1 minus straight n end fraction space tanα.$

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127.

Show that: $left parenthesis sin space 3 straight x space plus space sin space straight x right parenthesis space sin space straight x space plus space left parenthesis cos space 3 straight x space minus space cos space straight x right parenthesis space cosx space equals space 0$