What is expand 1/(sin(x)-1)+1/(1+sin(x)) ?

The solution to expand 1/(sin(x)-1)+1/(1+sin(x)) is -2sec(x)tan(x)

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expand 1/(sin(x)-1)+1/(1+sin(x))

expand

\frac{1}{sin ( x ) - 1} + \frac{1}{1 + sin ( x )}

- 2 sec (x) tan (x)

Solution steps

\frac{1}{sin ( x ) - 1} + \frac{1}{1 + sin ( x )}

Least Common Multiplier of

sin (x) - 1, 1 + sin (x) : (sin (x) - 1) (sin (x) + 1)

Adjust Fractions based on the LCM

= \frac{sin ( x ) + 1}{( sin ( x ) - 1 ) ( sin ( x ) + 1 )} + \frac{sin ( x ) - 1}{( sin ( x ) - 1 ) ( sin ( x ) + 1 )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{sin ( x ) + 1 + sin ( x ) - 1}{( sin ( x ) - 1 ) ( sin ( x ) + 1 )}

sin (x) + 1 + sin (x) - 1 = 2 sin (x)

= \frac{2 sin ( x )}{( sin ( x ) - 1 ) ( sin ( x ) + 1 )}

Expand

(- 1 + sin (x)) (1 + sin (x)) : sin^{2} (x) - 1

= \frac{2 sin ( x )}{- 1 + sin ^{2} ( x )}

Rewrite using trig identities

= - 2 sec (x) tan (x)

e^{x} (e^{x} - 2)

- 5^{n} + 7 (- 5^{n - 1})

\frac{d}{d t} ((C) (8000 + 2 x (C, t)))

(5 x^{2} + x + 1) e^{x}

(3 x + 2) (\frac{( 2 x - 3 ) ( 4 x - 2 )}{2})

What is expand 1/(sin(x)-1)+1/(1+sin(x)) ?
The solution to expand 1/(sin(x)-1)+1/(1+sin(x)) is -2sec(x)tan(x)