Question

Plz answer the above question with explanation ​

Answer 1

To find :

$\bold{ \: the \: value \: of \: \frac{se {c}^{2} \theta - cose {c}^{2} \theta }{se {c}^{2} \theta \: + cose {c}^{2} \theta } }$

Given :

$\bold{ cot \theta = \frac{1}{ \sqrt{7} }}$

Always Remember :

$\bull \: \bold{cose {c}^{2} \theta - {cot}^{2} \theta = 1} \\ \bullet \: \bold{ cot \theta = cosec \theta \times \frac{1}{sec \theta} }$

Solution :

as from the above identity , we know that :

$\bold{ cose {c}^{2} \theta - {cot}^{2} \theta = 1 } \\ \implies \bold{ {cosec}^{2} \theta - \frac{1}{{7} } = 1} \\ \implies \bold{ {cosec}^{2} \theta =1 + \frac{1}{ {7} } } \\ \implies \bold{ {cosec}^{2} \theta = \frac{8}{7} }$

hence, the value of cosec² theta:

$\bold{ {cosec}^{2} \theta = \frac{8}{7} } \\ \implies \bold{cosec \theta = \sqrt{ \frac{8}{7} } } \\ \implies \bold{ cosec \theta = \frac{ \sqrt{8} }{ \sqrt{7} } }$

From the second identity Above, we know that :

$\bold{cot \theta = cosec \theta \times \frac{1}{sec \theta} } \\ \implies \bold{ \frac{1}{ \sqrt{7} } = \frac{ \sqrt{8} }{ \sqrt{7} } \times \frac{1}{sec \: \theta} } \\ \implies \bold{sec \theta = \cancel{\sqrt{7} } \times \frac{ \sqrt{8} }{ \cancel{\sqrt{7}} } } \\ \implies \bold{sec \theta = \sqrt{8} }$

hence,

$\bold{ \frac{ {sec}^{2} \theta- {cosec}^{2} \theta }{ {sec}^{2} \theta + {cosec}^{2} \theta } } \\ \\ \implies \bold{ \frac{ {sec}^{2} \theta- {cosec}^{2} \theta }{ {sec}^{2} \theta + {cosec}^{2} \theta } } \bold{ = \frac{8 - \frac{8}{7}}{8 + \frac{8}{7} } } \\ \\ \implies \: \bold{ \frac{ {sec}^{2} \theta- {cosec}^{2} \theta }{ {sec}^{2} \theta + {cosec}^{2} \theta } } = \bold{ \frac{ \frac{ {48}}{ \cancel7}}{ \frac{64}{ \cancel7} } } \\ \\ \implies \bold{ \frac{ {sec}^{2} \theta- {cosec}^{2} \theta }{ {sec}^{2} \theta + {cosec}^{2} \theta } } = \bold{ \frac{ \cancel{48}}{ \cancel{64}} } \\ \\ \implies \: \bold{ \frac{ {sec}^{2} \theta- {cosec}^{2} \theta }{ {sec}^{2} \theta + {cosec}^{2} \theta } } = \bold{ \frac{2}{3} }$

hence, your answer is 2/3

Other Information:

$\bull \: \bold{sin \theta = \frac{1}{cosec \theta} } \\ \bull \: \bold{cos \theta = \frac{1}{sec \theta} } \\ \bull \: \bold{cot \theta = \frac{1}{tan \theta } } \\ \bull \: \bold{tan \theta = \frac{sin \theta}{cos \theta} } \\ \bull \: \bold{cot \theta = \frac{cos \theta}{sin \theta} }$