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Maths-

General

Easy

Question

If $x over y equals fraction numerator cos begin display style space end style A over denominator cos begin display style space end style B end fraction$ then $fraction numerator x tan space A plus y tan space B over denominator x plus y end fraction$ =

$\tan \frac{A + B}{2}$
$\tan \frac{A - B}{2}$
$\cot \frac{A + B}{2}$
$\cot \frac{A - B}{2}$

hint Hint:

Using sine and cosine trigonometry formula, solve the question.

The correct answer is: $tan space fraction numerator A plus B over denominator 2 end fraction$

$x over y equals fraction numerator cos A over denominator cos B end fraction space space space space space......... open parentheses 1 close parentheses U sin g space C o m p o n e n d o comma fraction numerator x plus y over denominator y end fraction equals fraction numerator cos A plus cos B over denominator cos B end fraction A p p l y i n g space f o r m u l a space c o s A plus cos B equals 2 cos open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses cos open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses fraction numerator x plus y over denominator y end fraction equals fraction numerator 2 cos open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses cos open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses over denominator cos B end fraction space space space........... open parentheses 2 close parentheses N o w comma fraction numerator x tan space A plus y tan space B over denominator x plus y end fraction D i v i d i n g space a l l space t h e space t e r m s space b y space y comma fraction numerator begin display style x over y end style tan space A plus tan space B over denominator begin display style fraction numerator x plus y over denominator y end fraction end style end fraction S u b s t i t u t i n g space e q u a t i o n space 1 space a n d space 2 space i n space a b o v e space e q u a t i o n comma equals fraction numerator begin display style fraction numerator cos A over denominator cos B end fraction tan space A plus tan space B end style over denominator begin display style fraction numerator 2 cos open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses cos open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses over denominator cos B end fraction end style end fraction equals fraction numerator begin display style fraction numerator cos A cross times sin A over denominator cos B cross times c o s A end fraction plus fraction numerator begin display style s i n space B end style over denominator cos B end fraction end style over denominator begin display style fraction numerator 2 cos open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses cos open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses over denominator cos B end fraction end style end fraction equals fraction numerator begin display style fraction numerator sin A over denominator cos B end fraction plus fraction numerator begin display style s i n space B end style over denominator cos B end fraction end style over denominator begin display style fraction numerator 2 cos open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses cos open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses over denominator cos B end fraction end style end fraction equals fraction numerator begin display style sin A plus sin B end style over denominator begin display style 2 cos open parentheses fraction numerator begin display style A plus B end style over denominator begin display style 2 end style end fraction close parentheses cos open parentheses fraction numerator begin display style A minus B end style over denominator begin display style 2 end style end fraction close parentheses end style end fraction A p p l y i n g space f o r m u l a space sin A plus sin B equals 2 sin open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses cos open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses equals fraction numerator begin display style 2 sin open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses cos open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses end style over denominator begin display style 2 cos open parentheses fraction numerator begin display style A plus B end style over denominator begin display style 2 end style end fraction close parentheses cos open parentheses fraction numerator begin display style A minus B end style over denominator begin display style 2 end style end fraction close parentheses end style end fraction equals fraction numerator sin open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses over denominator cos open parentheses fraction numerator begin display style A plus B end style over denominator begin display style 2 end style end fraction close parentheses end fraction equals tan open parentheses fraction numerator begin display style A plus B end style over denominator begin display style 2 end style end fraction close parentheses$

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