The answer to whether tan(x)cot(x)=1 is True

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prove tan(x)cot(x)=1

Solution

prove

tan (x) cot (x) = 1

True

Solution steps

tan (x) cot (x) = 1

Manipulating left side

tan (x) cot (x)

Express with sin, cos

cot (x) tan (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

\frac{cos ( x )}{sin ( x )} \cdot \frac{sin ( x )}{cos ( x )} = 1

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

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

Cancel the common factor:

sin (x)

= 1

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

p ro v e \frac{1}{2} (cot (x) + tan (x)) = csc (2 x)

p ro v e tan (- x) cos (x) = - sin (x)

p ro v e tan (2 π - x) = - tan (x)

p ro v e sec (2 x) = \frac{sec ^{2} ( x )}{2 - sec ^{2} ( x )}

p ro v e tan (a) + cot (a) = sec (a) csc (a)