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Trigonometric Functions

Short Answer Type

81.

If $tan left parenthesis straight alpha plus straight theta right parenthesis space equals space straight n space tan left parenthesis straight alpha minus straight theta right parenthesis comma$ show that $left parenthesis straight n plus 1 right parenthesis space sin 2 straight theta space equals space left parenthesis straight n minus 1 right parenthesis space sin 2 straight alpha$

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82.

Find the maximum and minimum values of:

$straight a space cosx plus straight b space sinx$

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83.

Find the maximum and minimum values of:

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84.

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85.

Prove the following:

$cos 4 straight x space equals space 1 minus 8 sin squared straight x space cos squared straight x$

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86.

Prove the following:

$tan 4 straight x space equals space fraction numerator 4 tanx space left parenthesis 1 minus tan squared straight x right parenthesis over denominator 1 minus 6 tan squared straight x plus tan to the power of 4 straight x end fraction$

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87.

Prove that:

$fraction numerator sin 2 straight x over denominator 1 plus cos 2 straight x end fraction space equals space tanx$

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88.

Prove that:

$fraction numerator 1 plus sin 2 straight x minus cos 2 straight x over denominator 1 plus sin 2 straight x plus cos 2 straight x end fraction space equals space tanx$

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89.
If $tan open parentheses straight x over 2 close parentheses space equals space 3 over 4 comma$ find the value of:

$sin space straight x$

We know that

   $sin space 2 straight A space equals space fraction numerator 2 space tanA over denominator 1 plus tan squared straight A end fraction$

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

   $sinx space equals space fraction numerator 2 tan open parentheses begin display style straight x over 2 end style close parentheses over denominator 1 plus tan squared open parentheses begin display style straight x over 2 end style close parentheses end fraction space equals space fraction numerator 2 open parentheses begin display style 3 over 4 end style close parentheses over denominator 1 plus open parentheses begin display style 3 over 4 end style close parentheses squared end fraction space equals space 3 over 2 cross times 16 over 25 equals space 24 over 25$

Hence,    $sinx space equals space 24 over 25$

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90.

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

$space space cos space straight x$