Is tan(θ)csc(θ)cos(θ)=1 ?

The answer to whether tan(θ)csc(θ)cos(θ)=1 is True

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prove tan(θ)csc(θ)cos(θ)=1

prove

tan (θ) csc (θ) cos (θ) = 1

True

Solution steps

tan (θ) csc (θ) cos (θ) = 1

Manipulating left side

tan (θ) csc (θ) cos (θ)

Express with sin, cos

cos (θ) csc (θ) tan (θ)

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

= cos (θ) \frac{1}{sin ( θ )} tan (θ)

Use the basic trigonometric identity:

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

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

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

cos (θ) \frac{1}{sin ( θ )} \cdot \frac{sin ( θ )}{cos ( θ )}

Multiply fractions:

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

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

Cancel the common factor:

sin (θ)

= \frac{1 \cdot cos ( θ )}{cos ( θ )}

Cancel the common factor:

cos (θ)

= 1

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

1 + tan^{2} (θ) = sec^{2} (θ)

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

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

cos (3 x) = cos^{3} (x) - 3 sin^{2} (x) cos (x)

tan (x) (1 + cos (2 x)) = sin (2 x)