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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:

$sin space straight x$

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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:

$space space cos space straight x$

We know that

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

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

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Hence, $cosx space equals space 7 over 25$