Is (sec(α)+csc(α))(cos(α)-sin(α))=cot(α)-tan(α) ?

The answer to whether (sec(α)+csc(α))(cos(α)-sin(α))=cot(α)-tan(α) is True

Popular >

prove (sec(α)+csc(α))(cos(α)-sin(α))=cot(α)-tan(α)

prove

(sec (α) + csc (α)) (cos (α) - sin (α)) = cot (α) - tan (α)

True

Solution steps

(sec (α) + csc (α)) (cos (α) - sin (α)) = cot (α) - tan (α)

Manipulating left side

(sec (α) + csc (α)) (cos (α) - sin (α))

Express with sin, cos

= \frac{( cos ( α ) + sin ( α ) ) ( cos ( α ) - sin ( α ) )}{cos ( α ) sin ( α )}

Expand

(cos (α) + sin (α)) (cos (α) - sin (α)) : cos^{2} (α) - sin^{2} (α)

= \frac{cos ^{2} ( α ) - sin ^{2} ( α )}{cos ( α ) sin ( α )}

Manipulating right side

cot (α) - tan (α)

Express with sin, cos

= \frac{cos ^{2} ( α ) - sin ^{2} ( α )}{cos ( α ) sin ( α )}

We showed that the two sides could take the same form

\Rightarrow True

\frac{tan ^{2} ( θ )}{sec ^{2} ( θ )} = (1)

1 - (- cos (x))^{2} = 1 + cos^{2} (x)

cos (x) = \frac{( csc ( x ) cot ( x ) )}{csc ^{2} ( x )}

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

\frac{sec ( x )}{- csc ( x )} + \frac{- sin ( x )}{cos ( x )} = - 2 tan (x)

Is (sec(α)+csc(α))(cos(α)-sin(α))=cot(α)-tan(α) ?
The answer to whether (sec(α)+csc(α))(cos(α)-sin(α))=cot(α)-tan(α) is True