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  • If tan alpha equals mdividemplus1 tan beta equals 1 divide 2mplus1 then alphaplusbeta equals A.pidivide2

If \tan \alpha=\frac{\mathrm{m}}{\mathrm{m}+1}, \tan \beta=\frac{1}{2 \mathrm{m}+1}  then \alpha + \beta is equal to

\\A. \frac{\pi}{2}\\\\ B.\frac{\pi}{3}\\\\ C.\frac{\pi}{c} \\\\ D.\frac{\pi}{4}

Answers (1)

The answer is the option (d).

\\ \tan \alpha =\frac{m}{m+1} \\\\ \tan \beta =\frac{m}{2m+1} \\\\ \tan \left( \alpha + \beta \right) =\frac{\tan \alpha +\tan \beta }{1 - \tan \alpha \tan \beta }\\\\=\frac{\frac{m}{m+1}+\frac{1}{2m+1}}{1 - \frac{m}{m+1} \times \frac{1}{2m+1}}\\\\=\frac{\frac{2m^{2}+m+m+1}{ \left( m+1 \right) \left( 2m+1 \right) }}{\frac{ \left( m+1 \right) \left( 2m+1 \right) - m}{ \left( m+1 \right) \left( 2m+1 \right) }}\\\\\\=\frac{2m^{2}+2m+1}{2m^{2}+2m+m+1 - m}\\\\=\frac{2m^{2}+2m+1}{2m^{2}+2m+1}\\\\=1 \\\\ \tan \left( \alpha + \beta \right) =\tan \frac{ \pi }{4} \\\\ \alpha + \beta =\frac{ \pi }{4} \\\\

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