Is (csc(x)+cot(x))(sec(x)-1)=tan(x) ?

The answer to whether (csc(x)+cot(x))(sec(x)-1)=tan(x) is True

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prove (csc(x)+cot(x))(sec(x)-1)=tan(x)

prove

(csc (x) + cot (x)) (sec (x) - 1) = tan (x)

True

Solution steps

(csc (x) + cot (x)) (sec (x) - 1) = tan (x)

Manipulating left side

(csc (x) + cot (x)) (sec (x) - 1)

Express with sin, cos

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

Rewrite using trig identities

= \frac{sin ( x )}{cos ( x )}

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

= tan (x)

We showed that the two sides could take the same form

\Rightarrow True

cos (x - \frac{π}{2}) = cos (x) tan (x)

csc (- θ) = - csc (θ)

cot^{2} (x) sin^{2} (x) = cos^{2} (x)

\frac{cos ( θ )}{tan ( θ ) + sec ( θ )} = 1 - sin (θ)

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

Is (csc(x)+cot(x))(sec(x)-1)=tan(x) ?
The answer to whether (csc(x)+cot(x))(sec(x)-1)=tan(x) is True