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

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prove tan(t)cot(t)=1

Solution

prove

tan (t) cot (t) = 1

True

Solution steps

tan (t) cot (t) = 1

Manipulating left side

tan (t) cot (t)

Express with sin, cos

cot (t) tan (t)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (t)

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

Cancel the common factor:

sin (t)

= 1

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

csc^{4} (θ) - cot^{4} (θ) = 2 csc^{2} (θ) - 1

csc (x) - cos (x) cot (x) = sin (x)

\frac{csc ^{2} ( θ )}{cot ( θ )} = csc (θ) sec (θ)

\frac{cot ^{3} ( t )}{csc ( t )} = cos (t) (csc^{2} (t) - 1)

sin (2 x) - tan (x) = tan (x) cos (2 x)