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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 $left parenthesis straight n plus 1 right parenthesis space sin 2 straight theta space equals space left parenthesis straight n minus 1 right parenthesis space sin 2 straight alpha$

tan space 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

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space space space space space fraction numerator sin space left parenthesis straight alpha plus straight theta right parenthesis over denominator cos space left parenthesis straight alpha plus straight theta right parenthesis end fraction space equals space fraction numerator nsin space left parenthesis straight alpha minus straight theta right parenthesis over denominator cos space left parenthesis straight alpha minus straight theta right parenthesis end fraction

Applying componendo and dividendo, we get

fraction numerator sin left parenthesis straight alpha plus straight theta right parenthesis space cos left parenthesis straight alpha minus straight theta right parenthesis space plus space cos left parenthesis straight alpha plus straight theta right parenthesis space sin left parenthesis straight alpha minus straight theta right parenthesis over denominator sin left parenthesis straight alpha plus straight theta right parenthesis space cos left parenthesis straight alpha minus straight theta right parenthesis space minus space cos left parenthesis straight alpha plus straight theta right parenthesis space sin left parenthesis straight alpha minus straight theta right parenthesis end fraction space equals space fraction numerator straight n plus 1 over denominator straight n minus 1 end fraction

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fraction numerator sin space left parenthesis straight alpha plus straight theta plus straight alpha minus straight theta right parenthesis over denominator sin space left parenthesis straight alpha plus straight theta minus straight alpha plus straight theta right parenthesis end fraction space equals space fraction numerator straight n plus 1 over denominator straight n minus 1 end fraction

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space space space fraction numerator sin 2 straight alpha over denominator sin 2 straight theta end fraction space equals space fraction numerator straight n plus 1 over denominator straight n minus 1 end fraction

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left parenthesis straight n minus 1 right parenthesis space sin 2 straight alpha space equals space left parenthesis straight n plus 1 right parenthesis space sin space 2 straight theta

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82.

Find the maximum and minimum values of:

$straight a space cosx plus straight b space sinx$

124 Views

83.

Find the maximum and minimum values of:

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84.

145 Views

85.

Prove the following:

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

101 Views

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:

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

121 Views

88.

Prove that:

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

106 Views

89.

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

$sin space straight x$