Question

Exercise 5.4/
If a + B = y = 180°, prove that the followings:
i. Tana + Tanß + Tany = Tana. Tanß. Tany
ii.
CotaCotß + CotßCoty + Coty Cota = 1
iii. Tan2a + Tan2B + Tan2y = Tan2a. Tan2ß. Tan2y
B
r
B В
iv.
+ Cot + Cot
Cot CotCot /
2
2​

Answer 1

Step-by-step explanation:

GIVEN :-

$\sf \alpha + \beta + \gamma = 180 \degree$

TO PROOF :-

$1). \sf \: \: \tan( \alpha ) + \tan( \beta ) + \tan( \gamma ) = \tan( \alpha ) .\tan( \beta ) . \tan( \gamma )$

$2). \: \: \sf \: \cot( \alpha ) . \cot( \beta ) + \cot( \beta ) . \cot( \gamma ) + \cot( \gamma ) . \cot( \alpha ) = 1$

SOLUTION :-

$\sf \: It \: \: is \: \: given \: \: that -$

$\sf \alpha + \beta + \gamma = 180 \degree$

$\implies \alpha + \beta = 180 - \gamma$

$\sf \: Then$

$\implies \tan(\alpha + \beta) = \tan(180 - \gamma )$

$\sf \: We \: \: know \: \: the \: \: formula -$

$\star \orange{ \sf \: \tan(180 - \theta) = - \tan( \theta) }$

$\star \orange{ \sf \: \tan(x + y) = \frac{ \tan(x) + \tan(y) }{1 - \tan(x). \tan(y) } }$

$\sf \: Now \: \: putting \: \: in \: \: above \: \: eq. -$

$\implies \tan(\alpha + \beta) = \tan(180 - \gamma )$

$\implies \sf \: \frac{ \tan( \alpha ) + \tan( \beta ) }{1 - \tan( \alpha ). \tan( \beta ) } = - \tan( \gamma )$

$\sf \: Now \: \: cross \: \: multiple$

$\implies \sf \tan( \alpha ) + \tan( \beta ) = - \tan( \gamma ) (1 - \tan( \alpha ) . \tan( \beta ) )$

$\sf \implies \: \tan( \alpha ) + \tan( \beta ) = - \tan( \gamma ) + \tan( \alpha ) \tan( \beta ) \tan( \gamma )$

$\sf \implies \: \tan( \alpha ) + \tan( \beta ) + \tan( \gamma ) = \tan( \alpha ) \tan( \beta ) \tan( \gamma )$

$\sf \: 1st. \: \: \: Proved -$

$\sf \green{\tan( \alpha ) + \tan( \beta ) + \tan( \gamma ) = \tan( \alpha ) \tan( \beta ) \tan( \gamma ) }$

$\sf \: Now \: \: second \: \: Prove -$

$\sf \: Divide \: by \: \: \: \: \tan( \alpha ) \tan( \beta ) \tan( \gamma ) \: both \: \: sides$

$\sf \: in \: first \: \: prove$

$\sf \: \frac{ \tan( \alpha ) + \tan( \beta ) + \tan( \gamma ) }{ \tan( \alpha ) \tan( \beta ) \tan( \gamma ) } = \frac{ \tan( \alpha ) \tan( \beta ) \tan( \beta ) }{ \tan( \alpha ) \tan( \beta ) \tan( \gamma ) }$

$\implies \sf \: \frac{1}{ \tan( \beta ) \tan( \gamma ) } + \frac{1}{ \tan( \alpha ) \tan( \gamma ) } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: + \frac{1}{ \tan( \alpha ) \tan( \beta ) } = 1$

$\implies \sf \: \frac{1}{ \tan( \beta ) } . \frac{1}{ \tan( \gamma ) } + \frac{1}{ \tan( \alpha ) } . \frac{1}{ \tan( \gamma ) } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: + \frac{1}{ \tan( \alpha ) } . \frac{1}{ \tan( \beta ) } = 1$

$\sf \: Formula -$

$\orange{ \sf \frac{1}{ \tan( x) } = \cot(x)}$

$\implies \sf \: \cot( \beta ) . \cot( \gamma ) + \cot( \alpha ) \cot( \gamma ) \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: + \cot( \alpha ) \cot( \beta ) = 1$

$\implies \sf \: \cot( \alpha ) . \cot( \beta ) + \cot( \beta ) . \cot( \gamma ) \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: + \cot( \gamma ) . \cot( \alpha ) = 1$

$\sf \: Hence \: \: second \: \: proved -$

$\green{ \sf \cot( \alpha ) . \cot( \beta ) + \cot( \beta ) . \cot( \gamma ) } \\ \green{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: + \cot( \gamma ) . \cot( \alpha ) = 1}$

$\sf \bold{ \sf \: SIMILARLY \: \: FOR \: \: 3rd \: \: PROVING - }$

$\sf \: It \: \: is \: same \: as \: 1st \: proving \: just \: multiplying \: by \: 2$

$\sf \alpha + \beta + \gamma = 180 \degree$

$\sf 2(\alpha + \beta + \gamma) = 2(180 \degree )$

$2 \alpha + 2 \beta = 360 - 2 \gamma$

$\implies \sf \: \: \tan(2 \alpha + 2 \beta ) = \tan(360 - 2 \gamma )$

$\sf \: NOW \: \: solve \: \: it \: \: as \: \: first \: \: we \: \: get$

$\sf \green{\tan( 2\alpha ) + \tan( 2\beta ) + \tan(2 \gamma )} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \green{ = \tan( 2\alpha ) \tan( 2\beta ) \tan( 2\gamma ) }$