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Trigonometric Functions

Short Answer Type

571.

Prove that:

$sin open parentheses straight x over 2 close parentheses space equals space plus-or-minus square root of fraction numerator 1 minus cosx over denominator 2 end fraction end root$

103 Views

572.
Prove that:

$tan open parentheses straight x over 2 close parentheses equals plus-or-minus square root of fraction numerator 1 minus cosx over denominator 1 plus cosx end fraction end root.$

We know that:
$cos space 2 straight A space equals space fraction numerator 1 minus tan squared straight A over denominator 1 plus tan squared straight A end fraction$

Replacing A by $space space straight x over 2 comma$ we get

 $cosx space equals space fraction numerator 1 minus tan squared open parentheses begin display style straight x over 2 end style close parentheses over denominator 1 plus tan squared open parentheses begin display style straight x over 2 end style close parentheses end fraction$
$rightwards double arrow$ $fraction numerator 1 plus tan squared open parentheses begin display style straight x over 2 end style close parentheses over denominator 1 minus tan squared open parentheses begin display style straight x over 2 end style close parentheses end fraction equals 1 over cosx$
Apply componendo and dividendo, we get

$rightwards double arrow$ $fraction numerator 1 plus tan squared open parentheses begin display style straight x over 2 end style close parentheses plus 1 minus tan squared open parentheses begin display style straight x over 2 end style close parentheses over denominator 1 plus tan squared open parentheses begin display style straight x over 2 end style close parentheses minus 1 plus tan squared open parentheses begin display style straight x over 2 end style close parentheses end fraction equals fraction numerator 1 plus cosx over denominator 1 minus cosx end fraction$ $rightwards double arrow$ $fraction numerator 2 over denominator 2 space tan squared open parentheses begin display style straight x over 2 end style close parentheses end fraction equals fraction numerator 1 plus cosx over denominator 1 minus cosx end fraction$

$rightwards double arrow$ $space space fraction numerator 1 over denominator tan squared open parentheses begin display style straight x over 2 end style close parentheses end fraction equals fraction numerator 1 plus cosx over denominator 1 minus cosx end fraction$ $rightwards double arrow$ $tan squared open parentheses straight x over 2 close parentheses space equals space fraction numerator 1 minus cosx over denominator 1 plus cosx end fraction$

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210 Views

573.

Prove that:

$2 space sinA space cosB space equals space sin left parenthesis straight A plus straight B right parenthesis space plus space sin left parenthesis straight A minus straight B right parenthesis$

101 Views

574.

Prove that:
2 cosA sinB = sin (A + B) - sin (A - B)

130 Views

575.

Prove that:
2 cosA cosB = cos (A + B) + cos (A - B)

113 Views

576.

Prove that:
-2 sinA sinB = cos (A + B) - cos (A - B)

218 Views

577.

Prove the following:

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

91 Views

578.

Prove the following:

$sin space straight C space minus space sin space straight D space equals space 2 cos fraction numerator straight C plus straight D over denominator 2 end fraction sin fraction numerator straight C minus straight D over denominator 2 end fraction$

100 Views

579.

Prove the following:

$space space cosC space plus space cosD space equals space 2 cos fraction numerator straight C plus straight D over denominator 2 end fraction cos fraction numerator straight C minus straight D over denominator 2 end fraction$

116 Views

580.

Prove the following:

$cosC space minus space cosD space equals space minus 2 sin fraction numerator straight C plus straight D over denominator 2 end fraction sin fraction numerator straight C minus straight D over denominator 2 end fraction$

90 Views

Previous Year Papers

Test Series

Test Yourself

Short Answer Type

572. Prove that:

572.
Prove that:

$tan open parentheses straight x over 2 close parentheses equals plus-or-minus square root of fraction numerator 1 minus cosx over denominator 1 plus cosx end fraction end root.$