Question

if cotθ = 7/8 evaluate
1) ((1 + sinθ)( 1 - sinθ)) / ((1 + sinθ)(1 - sinθ))

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Answer 1

Correct Question

$\large \sf{if \: \cotθ = \frac{7}{8} \: \: then \: evaluate: } \\ \\ \large \sf{1) \large\frac{(1 + \sinθ)(1 - \sinθ) }{(1 + \cosθ)(1 - \cosθ)} }$

Answer

First we will find Hypotenuse.

$\implies \large \cal{ \:H = \sqrt{(8k) ^{2} - (7k)^{2} } } \\ \\\implies \large \cal{ \: H = \sqrt{64k ^{2} - 49k^{2} } } \\ \\\implies \large \cal{ \: H= \sqrt{113k^{2} } } \\ \\ \implies \large \cal \red {\boxed{ \: H= \sqrt{113} \: {k } }}$

Now we will find sinθ and cosθ

$\implies \large \cal{ \sinθ = {\large\frac{P}{H}} = {\large\frac{8k}{ \sqrt{ 113} \: k}}\implies \large \red{ \frac{5}{ \sqrt{ 113}} }} \\ \\ \implies \large \cal{ \cosθ = {\large\frac{B}{H}} = {\large\frac{7k}{ \sqrt{ 113} \: k}}\implies \large\red{ \frac{7}{ \sqrt{ 113}} }}$

Now we will use this formula a² - b² =(a + b)(a - b)

$\implies\large\frac{ \large(1)^{2} - ({\sin^{2} }{θ})}{ \large(1)^{2} - ({\cos}^{2} {θ}) } \\ \\ \implies\large\frac{ \large1 - ({\sin^{2} }{θ})}{ \large1- ({\cos}^{2} {θ}) }$

Now put values in the formula.

$\implies \frac{ \large{1 - ( \frac{8}{ \sqrt{113} })^{2} }}{ \large{1 - (\frac{7}{ \sqrt{113} })^{2}}} \\ \\ \implies \frac{ \large{1 - \frac{64}{113}}}{ \large{1 - \frac{49}{{113} }}}$

$\implies \frac{ \large{1 - \frac{64}{ {113}}}}{ \large{1 - \frac{49}{{ {113}} }}} \\ \\ \implies \frac{ \large{ \frac{49}{ { \cancel{113}}}}}{ \large{ \frac{64}{{ { \cancel{113}}} }}} \\ \\ \implies \red{\boxed{ \frac{ \large 49}{ \large{64} }} \: \: \: \: \: Ans}$