Question

if a and b are acute angle such that

tan a = m / m+1 and tan b = 1/ 2m +1 , prove that a + b = π/4 .​

Answer 1

รσℓµƭเσɳ

tan ( a+b ) = tan a + tan b/ 1- tan a tan b

Putting the value of tan a and tan b .

tan ( a+b ) =( m / m+1 + 1/ 2m + 1 )/ 1 - (m/ m+1 )(1/2 m+1)

tan ( a+b ) = 2m² +m +m +1/ 2m² + 3m +1 - m

tan (a+b ) = 2m² +2m +1/ 2m²+2m +1

tan ( a+b) = 1

tan ( a+b) = tan π/4

a+b = π/4

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Answer 2

Step-by-step explanation:

$\tan( \alpha ) = \frac{x}{x + 1} \\ \\ \\ \tan( \beta ) = \frac{1}{2x + 1} \\ \\ \\ \tan( \alpha + \beta ) = \frac{ \tan( \alpha ) + \tan( \beta ) }{1 - \tan( \alpha ) \tan( \beta ) } \\ \\ \\ = \frac{ \frac{x}{x + 1} + \frac{1}{2x + 1} }{1 - \frac{x}{(x + 1)(2x + 1)} } \\ \\ \\ = \frac{ \frac{2 {x}^{2} + x + x + 1 }{(x + 1)(2x + 1)} }{ \frac{ 2{x}^{2} + 2x + x + 1 - x}{(x + 1)(2x + 1)} } \\ \\ \\ = \frac{2 {x}^{2} + 2x + 1}{2 {x}^{2} + 2x + 1} \\ \\ \\ = 1 \\ \\ \\ but \: \tan( \frac{\pi}{4} ) = 1 \\ \\ \\ \alpha + \beta = \frac{\pi}{4}$

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