prove cos(a)sec(a)=1

Solution

prove

cos (a) sec (a) = 1

True

Solution steps

cos (a) sec (a) = 1

Manipulating left side

cos (a) sec (a)

sec (a) cos (a) = 1

sec (a) cos (a)

Express with sin, cos

sec (a) cos (a)

Use the basic trigonometric identity:

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

= \frac{1}{cos ( a )} cos (a)

\frac{1}{cos ( a )} cos (a) = 1

\frac{1}{cos ( a )} cos (a)

Multiply fractions:

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

= \frac{1 cos ( a )}{cos ( a )}

Cancel the common factor:

cos (a)

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

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

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

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

cos (3 π) = - 1

- sec^{2} (x) sin^{2} (x) = 1 - sec^{2} (x)