Question

$(cosec\theta -sin\theta )(sec\theta -cos\theta )(tan\theta +cot\theta )=1$

Answer 1

$LHS=(cosec\theta -sin\theta )(sec\theta -cos\theta )(tan\theta +cot\theta )$

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

$(\frac{sin\theta }{cos\theta }+\frac{cos\theta }{sin\theta })$

$\left[\begin{array}{c}\because cosec\theta =\frac{1}{sin\theta },sec\theta =\frac{1}{cos\theta },\\ tan\theta =\frac{sin\theta }{cos\theta }andcot\theta =\frac{cos\theta }{sin\theta }\end{array}\right]$

$=\left(\frac{1-si{n}^2\theta }{sin\theta }\right)\left(\frac{1-co{s}^2\theta }{cos\theta }\right)$

$\left(\frac{si{n}^2\theta +co{s}^2\theta }{cos\theta sin\theta }\right)$

$=\frac{co{s}^2\theta .si{n}^2.1}{si{n}^2\theta .co{s}^2\theta }$

$\left[\begin{array}{c}\because 1-si{n}^2\theta =co{s}^2\theta ,and\\ 1-co{s}^2\theta =si{n}^2\theta \end{array}\right]$

=1=RHS

Hence proved.