Question

if cot teta= 5/12 then find 1+ sin teta 1- sin teta​

Answer 1

$\huge\purple{\underline{\underline{\pink{Ans}\red{wer:-}}}}$

$\sf{The \ value \ of \ (1+sin\theta)(1-sin\theta) \ is \ \frac{-25}{169}}$

$\sf\orange{Given:}$

$\sf{\implies{cot\theta=\frac{5}{12}}}$

$\sf\pink{To \ find:}$

$\sf{(1+sin\theta)(1-sin\theta)}$

$\sf\green{\underline{\underline{Solution:}}}$

$\sf{\implies{cot\theta=\frac{5}{12}}}$

$\sf{cosec^{2}\theta-cot^{2}\theta=1}$

$\sf{... Trignometric \ identity}$

$\sf{\implies{cosec^{2}\theta-(\frac{5}{12})^{2}=1}}$

$\sf{\implies{cosec^{2}\theta=1+\frac{25}{144}}}$

$\sf{\implies{cosec^{2}\theta=\frac{144+25}{144}}}$

$\sf{\implies{cosec^{2}\theta=\frac{169}{144}}}$

$\sf{On \ taking \ square \ root \ of \ both \ sides}$

$\sf{\implies{cosec\theta=\frac{13}{12}}}$

$\sf{cosec\theta=\frac{1}{sin\theta}}$

$\sf{... Trignometric \ identity}$

$\sf{\implies{\therefore{sin\theta=\frac{12}{13}}}}$

____________________________________

$\sf{\implies{(1+sin\theta)(1-sin\theta)}}$

$\sf{By \ identity}$

$\sf{a^{2}-b^{2}=(a+b)(a-b)}$

$\sf{\implies{\therefore{1-sin^{2}\theta}}}$

$\sf{\implies{1-sin^{2}\theta}}$

$\sf{\implies{1-(\frac{12}{13})^{2}}}$

$\sf{\implies{1-\frac{144}{169}}}$

$\sf{\implies{\frac{169-144}{169}}}$

$\sf{\implies{\frac{-25}{169}}}$

$\sf\purple{\tt{\therefore{The \ value \ of \ (1+sin\theta)(1-sin\theta) \ is \ \frac{-25}{169}}}}$