Prove this
- Thread starter Drizzt
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- Feb 4, 2004
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Assuming you mean:
. . . . .\(\displaystyle \large{\frac{1}{1\,-\,\sec{(x)}}\;+\;\frac{1}{1\,+\,\sec{(x)}}\;=\;-2\cot^2{(x)}}\)
...a good first step might be to convert to a common denominator on the left, and combine the fractions. See where this takes you.
Eliz.
. . . . .\(\displaystyle \large{\frac{1}{1\,-\,\sec{(x)}}\;+\;\frac{1}{1\,+\,\sec{(x)}}\;=\;-2\cot^2{(x)}}\)
...a good first step might be to convert to a common denominator on the left, and combine the fractions. See where this takes you.
Eliz.
There are always different approaches at trig identities, but you could try
\(\displaystyle \frac{1}{1-sec(x)}+\frac{1}{1+sec(x)}\)
\(\displaystyle \frac{1+sec(x)+1-sec(x)}{(1-sec(x))(1+sec(x))}\)
\(\displaystyle \frac{2}{(1-sec(x))(1+sec(x))}\)
\(\displaystyle \frac{2}{1-sec^{2}(x)}\)
\(\displaystyle \frac{-2}{sec^{2}(x)-1}\)
Since \(\displaystyle sec^{2}(x)-1=tan^{2}(x)\), we have
\(\displaystyle \frac{-2}{tan^{2}(x)}=-2cot^{2}(x)\)
\(\displaystyle \frac{1}{1-sec(x)}+\frac{1}{1+sec(x)}\)
\(\displaystyle \frac{1+sec(x)+1-sec(x)}{(1-sec(x))(1+sec(x))}\)
\(\displaystyle \frac{2}{(1-sec(x))(1+sec(x))}\)
\(\displaystyle \frac{2}{1-sec^{2}(x)}\)
\(\displaystyle \frac{-2}{sec^{2}(x)-1}\)
Since \(\displaystyle sec^{2}(x)-1=tan^{2}(x)\), we have
\(\displaystyle \frac{-2}{tan^{2}(x)}=-2cot^{2}(x)\)