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Trigonometric Functions

Short Answer Type

91.

If $tan open parentheses straight x over 2 close parentheses space equals space 3 over 4 comma$ find the value of:

tan x.

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92.

Show that: $space space space square root of 2 plus square root of 2 plus 2 space cos space 4 straight theta end root end root space equals space 2 cosθ$

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93. $If space space tanx space equals space straight b over straight a comma$ find the value of $straight a space cos space 2 straight x space plus space straight b space sin space 2 straight x.$

Using    $space space cos space 2 straight x space equals space fraction numerator 1 minus tan squared straight x over denominator 1 plus tan squared straight x end fraction$ and

   $sin 2 straight x space equals space fraction numerator 2 tanx over denominator 1 plus tan squared straight x end fraction comma$ we get

$space space space straight a space cos space 2 straight x plus straight b space sin space 2 straight x equals straight a open square brackets fraction numerator 1 minus tan squared straight x over denominator 1 plus tan squared straight x end fraction close square brackets plus straight b open square brackets fraction numerator 2 tanx over denominator 1 plus tan squared straight x end fraction close square brackets space equals space straight a open square brackets fraction numerator 1 minus begin display style straight b squared over straight a squared end style over denominator 1 plus begin display style straight b squared over straight a squared end style end fraction close square brackets plus straight b open square brackets fraction numerator 2 open parentheses begin display style straight b over straight a end style close parentheses over denominator 1 plus begin display style straight b squared over straight a squared end style end fraction close square brackets$

   $space space space equals straight a open square brackets fraction numerator straight a squared minus straight b squared over denominator straight a squared plus straight b squared end fraction close square brackets plus straight b open square brackets fraction numerator 2 ab over denominator straight a squared plus straight b squared end fraction close square brackets space equals space fraction numerator 1 over denominator straight a squared plus straight b squared end fraction left square bracket straight a cubed minus ab squared plus 2 ab squared right square bracket$

$equals fraction numerator 1 over denominator straight a squared plus straight b squared end fraction left square bracket straight a cubed plus ab squared right square bracket space equals space fraction numerator 1 over denominator straight a squared plus straight b squared end fraction cross times space straight a space left parenthesis straight a squared plus straight b squared right parenthesis space equals space straight a$

Hence,    $straight a space cos 2 straight x space plus space straight b space sin space 2 straight x space equals space straight a.$

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94.

Show that:

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95.

Show that:

$8 cos cubed open parentheses straight pi over 9 close parentheses minus 6 cos open parentheses straight pi over 9 close parentheses space equals space 1$

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96.

Show that:

$cos 4 straight A space equals space 1 minus 8 cos squared straight A space plus space 8 cos to the power of 4 straight A$

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97.

Show that:

$cos space 6 straight A space equals space 32 space cos to the power of 6 straight A space minus space 48 space cos to the power of 4 straight A space plus space 18 space cos squared straight A space minus 1$