Question

If cot $\theta =\frac{7}{8}$,evaluate:

(i) $\frac{(1+sin\theta )(1-sin\theta )}{(1+cos\theta )(1-cos\theta )}$

(ii) $co{t}^2\theta$

Answer 1

$cot\theta =\frac{7}{8}$ i.e., [$\frac{Base}{perpendicular}$]
Now, by Phythagoras Theorem
H = $\sqrt{64+49}$
H = $\sqrt{113}$
$\therefore sin\theta =\frac{8}{\sqrt{113}}$ and
$cos\theta =\frac{7}{\sqrt{113}}$

(i) $\frac{(1+sin\theta )(1-sin\theta )}{(1+cos\theta )(1-cos\theta )}$
$=\frac{1^2-(\frac{8}{\sqrt{113}}{)}^2}{1^2-(\frac{7}{\sqrt{113}}{)}^2}$ [putting values of sin​ θ and cos ​θ and using identitiy $a^2-b^2$]
$=\frac{1-\left(\frac{64}{113}\right)}{1-\left(\frac{49}{113}\right)}$
$=\frac{49}{64}$

(ii) $co{t}^2\theta =(\frac{7}{8}{)}^2$
$=\frac{49}{64}$