First slide

Trigonometric Identities

Question

If $a \sin^{2} x + b \cos^{2} x = c, b \sin^{2} y + a \cos^{2} y = d$ and $a \tan x = b \tan y$ then $\frac{a^{2}}{b^{2}} =$

Moderate

A

$\frac{(a - d) (c - a)}{(b - c) (d - b)}$
B

$\frac{(b - c) (b - d)}{(a - c) (a - d)}$
C

$\frac{(b - c) (d - b)}{(a - d) (c - a)}$
D

$\frac{(d - a) (c - a)}{(b - c) (d - b)}$

Solution

$a \sin^{2} x + b \cos^{2} x = c d e v i d e w i t h \cos^{2} x \Rightarrow a \tan^{2} x + b = c \sec^{2} x$
$\Rightarrow (a - c) \tan^{2} x = c - b \Rightarrow \tan^{2} x = \frac{c - b}{a - c}$
similarly $\tan^{2} y = \frac{d - a}{b - d}$ but $a \tan x = b \tan y \Rightarrow \frac{a^{2}}{b^{2}} = \frac{\tan^{2} y}{\tan^{2} x}$

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