if m tan(A-30)=n tan(A+120) then m+n/m-n=

Open in App

Solution

Dear student,
It is given that $m \tan (A - 30) = n \tan (A + 120)$
$\Rightarrow$ $\frac{m}{n} = \frac{\tan (A + 120)}{\tan (A - 30)}$
converting it into sin and cos ,we get
$\frac{m}{n} = \frac{\sin (A + 120) \cos (A - 30)}{\cos (A + 120) \sin (A - 30)}$
(as we know if $\frac{a}{b} = \frac{c}{d}$ then from componendo and dividendo $\frac{a + b}{a - b} = \frac{c + d}{c - d}$ )
so by applying componendo and dividendo,we get
$\Rightarrow$ $\frac{m + n}{m - n} = \frac{\sin (A + 120) \cos (A - 30) + \cos (A + 120) \sin (A - 30)}{\sin (A + 120) \cos (A - 30) - \cos (A + 120) \sin (A - 30)}$
by using formulas $\sin (A + B) = \sin A \cos B + \cos A \sin B$ and $\sin (A - B) = \sin A \cos B - \cos A \sin B$ we get
$\Rightarrow$ $\frac{m + n}{m - n} = \frac{\sin (A + 120 + A - 30)}{\sin (A + 120 - A + 30)}$
$\Rightarrow$ $\frac{m + n}{m - n} = \frac{\sin (90 + 2 A)}{\sin (150)}$ (as $\sin (90 + θ) = \cos θ$ )
$\Rightarrow$ $\frac{m + n}{m - n} = \frac{\cos (2 A)}{\frac{1}{2}}$
$\Rightarrow$ $\frac{m + n}{m - n} = 2 \cos (2 A)$
hope you understand
enjoy

Suggest Corrections

Similar questions

If $\frac{m}{n} = \frac{t a n A}{t a n B}$ , find the value of $\frac{m + n}{m - n}$ is

tan A + sin A = m

tan A + sin A = n

m^{2} - n^{2} = 3 \sqrt{m n}

tan A + sin A = m

tan A - sin A = n

m^{2} - n^{2} = 4 \sqrt{m n}

If $sec A + tan A = m$ and $sec A - tan A = n$ . Find the
value of $\sqrt{m n}$ .

Join BYJU'S Learning Program