Question

Refer to the attachment. Thank you!​

Answer 1

Answer:

x

Step-by-step explanation:

tan + cot = x

$\frac{ \sin}{ \cos } + \frac{ \cos }{ \sin } = x \\ \frac{ {sin}^{2} \: + {cos}^{2} }{sin \times cos } = x \\ \frac{1}{sin \times cos} = x \\ sec \times cosec = x$

Answer 2

$\huge{\underline{\underline{\green{\mathfrak{Answer:}}}}}$

Given

$\mathsf{tan\theta + cot\theta = x}$

To find

$\mathsf{sec\theta × cosec\theta}$

METHOD 1

1. Finding the value of cosecθ

$\mathsf{\frac{1}{cot\theta} + cot\theta = x}$

$\mathsf{\frac{1}{cot\theta} + cot\theta = x}$

$\mathsf{\frac{1 + cot^2\theta}{cot\theta} = x}$

$\mathsf{\frac{cosec^2\theta}{cot\theta} = x}$

$\mathsf{cosec^2\theta = x\:cot\theta}$

$\mathsf{cosec^2\theta = \sqrt{x\:cot\theta}\rightarrow\:(1)}$

2. Finding the value of secθ

$\mathsf{tan\theta + \frac{1}{tan\theta} = x}$

$\mathsf{\frac{tan^2 + 1}{tan\theta} = x}$

$\mathsf{\frac{sec^2\theta}{tan\theta} = x}$

$\mathsf{sec^2\theta = x\:tan\theta}$

$\mathsf{sec\theta = \sqrt{x\:tan\theta}\rightarrow\:(2)}$

Putting the values of cosecθ and secθ

$\mathsf{\implies\:sec\theta × cosec\theta}$

$\mathsf{\implies\:\sqrt{x\:tan\theta} × \sqrt{x\:cot\theta}}$

$\mathsf{\implies\:\sqrt{x}\:\sqrt{tan\theta} × \sqrt{x}\:\sqrt{cot\theta}}$

$\mathsf{\implies\:x\:\sqrt{tan\theta} × \sqrt{\frac{1}{tan\theta}}}$

$\mathsf{\implies\:x\:\sqrt{tan\theta} × \frac{1}{\sqrt{tan\theta}}}$

$\mathsf{Cancelling \:\sqrt{tan\theta}}$

$\fbox{\mathsf{\implies\:x}}$

METHOD 2

$\mathsf{\implies\:tan\theta + cot\theta = x}$

$\mathsf{\implies\: \frac{sin\theta}{cos\theta} + \frac{cos\theta}{sin\theta} = x}$

Taking LCM of the denominator

$\mathsf{\implies \: \frac{sin^2\theta + cos^2\theta}{cos\theta\:sin\theta} = x}$

$\fbox{\mathsf{Using \:identity\:: sin^2\theta + cos^2\theta = 1}}$

$\mathsf{\implies\: \frac{1}{cos\theta\:sin\theta} = x}$

$\mathsf{\implies\:\frac{1}{cos\theta} × \frac{1}{sin\theta} = x}$

$\fbox{\mathsf{Using\:inverse\:identity}}$

$\mathsf{\implies \:\sec\theta × cosec\theta = x}$