If-a-sin-2-x-b-cos-2-x-c-b-sin-2-y-a-cos-2-y-d-and-a-tan-2-x-b-tan-y-then-a-2-b-2-is-equal-to-

Question Number 18702 by Christo01 last updated on 28/Jul/17

If a \sin^{2} x + b \cos^{2} x = c,

b \sin^{2} y + a \cos^{2} y = d and

a \tan^{2} x = b \tan y

then \frac{a^{2}}{b^{2}} is equal to

Commented by mondodotto@gmail.com last updated on 29/Jul/17

please help me to solve this question

Answered by behi.8.3.4.1.7@gmail.com last updated on 30/Jul/17

a . {t g}^{2} x + b = c + c . {t g}^{2} x \Rightarrow {t g}^{2} x = \frac{c - b}{a - c}

b . {t g}^{2} y + a = d + d . {t g}^{2} y \Rightarrow {t g}^{2} y = \frac{d - a}{b - d}

a . {t g}^{2} x = b . {t g}^{2} y \Rightarrow a . \frac{c - b}{a - c} = b . \frac{d - a}{b - d} \Rightarrow

\Rightarrow a (b c - d c - b^{2} + b d) = b (a d - a^{2} - d c + a c)

a b c - a d c - {a b}^{2} + a b d = a b d - a^{2} b - b c d + a b c

a^{2} b - {a b}^{2} = a d c - b d c

a b (a - b) = c d (a - b) \Rightarrow {\begin{cases} a = b \Rightarrow \frac{a^{2}}{b^{2}} = 1 \\ a b = c d \end{cases}