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Trigonometric Functions

Short Answer Type

121.

Prove the following:

$space space space cot space 4 straight x space left parenthesis sin space 5 straight x space plus space sin space 3 straight x right parenthesis space equals space cot space straight x space left parenthesis sin space 5 straight x space minus space sin space 3 straight x right parenthesis$

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122.

Prove the following:

$sin space 2 straight x space plus space 2 space sin space 4 straight x space plus space sin space 6 straight x space equals space 4 cos squared space straight x space sin space 4 straight x$

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123.

Prove that:

$2 space cos straight pi over 13 space cos fraction numerator 9 straight pi over denominator 13 end fraction plus cos fraction numerator 3 straight pi over denominator 13 end fraction plus cos fraction numerator 5 straight pi over denominator 13 end fraction space equals space 0$

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124.

Prove that:

$cosα space plus space cosβ space plus space cosγ space plus space cos space left parenthesis straight alpha space plus space straight beta space plus space straight gamma right parenthesis space equals space 4 cos open parentheses fraction numerator straight alpha plus straight beta over denominator 2 end fraction close parentheses space cos open parentheses fraction numerator straight beta plus straight gamma over denominator 2 end fraction close parentheses space cos fraction numerator straight gamma plus straight alpha over denominator 2 end fraction.$

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125.

If cos (A + B) sin (C - D) = cos (A - B) sin (C + D)

Show that: tan A tan B tan C + tan D = 0

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126.

If $sin space straight theta space equals space straight n space sin space left parenthesis straight theta space plus space 2 straight alpha right parenthesis comma$ show that $tan space left parenthesis straight theta space plus space straight alpha right parenthesis space equals space fraction numerator 1 plus straight n over denominator 1 minus straight n end fraction space tanα.$

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127.
Show that: $left parenthesis sin space 3 straight x space plus space sin space straight x right parenthesis space sin space straight x space plus space left parenthesis cos space 3 straight x space minus space cos space straight x right parenthesis space cosx space equals space 0$

L.H.S. = $left parenthesis sin space 3 straight x space plus space sin space straight x right parenthesis space sin space straight x space plus space left parenthesis cos space 3 straight x space minus space cos space straight x right parenthesis space cos space straight x$

= $open square brackets 2 space sin open parentheses fraction numerator 3 straight x plus straight x over denominator 2 end fraction close parentheses space cos open parentheses fraction numerator 3 straight x minus straight x over denominator 2 end fraction close parentheses close square brackets sinx space plus space open square brackets negative 2 sin open parentheses fraction numerator 3 straight x plus straight x over denominator 2 end fraction close parentheses space sin open parentheses fraction numerator 3 straight x minus straight x over denominator 2 end fraction close parentheses close square brackets cosx$

$space space space space space space equals space 2 space sin 2 straight x space cosx space sinx space minus space 2 space sin space 2 straight x space sinx space cosx space equals space 0 space equals space straight R. straight H. straight S.$

Hence, L.H.S. = R.H.S.