What is the limit as x approaches x/2 of ((sin(x)-1))/(cos(x)) ?

The limit as x approaches x/2 of ((sin(x)-1))/(cos(x)) is sec(x/2)(sin(x/2)-1)

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limit as x approaches x/2 of ((sin(x)-1))/(cos(x))

x \to \frac{x}{2} lim (\frac{( sin ( x ) - 1 )}{cos ( x )})

sec (\frac{x}{2}) (sin (\frac{x}{2}) - 1)

Solution steps

x \to \frac{x}{2} lim (\frac{sin ( x ) - 1}{cos ( x )})

Plug in the value

x = \frac{x}{2}

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

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

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

What is the limit as x approaches x/2 of ((sin(x)-1))/(cos(x)) ?
The limit as x approaches x/2 of ((sin(x)-1))/(cos(x)) is sec(x/2)(sin(x/2)-1)