Q)Write the following expression in terms of cosine both in a simplified form?
$\displaystyle \frac{1-csc^2x}{csc^2x} $
Attempt- Is my attempt Correct )
$\displaystyle -1(sin^2x-1) =-cos^2x $
Thank you
$\displaystyle csc^2(x) = \frac{1}{sin^2(x)}$
$\displaystyle
1- \frac{1}{sin^2(x)} = \frac{sin^2(x) - 1}{sin^2(x)}$
Overall the sum becomes:
$\displaystyle \frac{sin^2(x) - 1}{sin^2(x)csc(x)}$ and so the denominator will be sin(x)
$\displaystyle sin^2(x) - 1 = -cos^2(x) and sin(x) = \sqrt{1-cos^2(x)}$
therefore we have $\displaystyle \frac{-cos^2(x)}{\sqrt{1-cos^2(x)}}$
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From your expression you can say that $\displaystyle -1(sin^2(x)-1) = 1-sin^2(x) = cos^2(x)$