Is 2cos(θ)tan(θ)csc(θ)=2 ?

The answer to whether 2cos(θ)tan(θ)csc(θ)=2 is True

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prove 2cos(θ)tan(θ)csc(θ)=2

prove

2 cos (θ) tan (θ) csc (θ) = 2

True

Solution steps

2 cos (θ) tan (θ) csc (θ) = 2

Manipulating left side

2 cos (θ) tan (θ) csc (θ)

Express with sin, cos

2 cos (θ) csc (θ) tan (θ)

Use the basic trigonometric identity:

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

= 2 cos (θ) \frac{1}{sin ( θ )} tan (θ)

Use the basic trigonometric identity:

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

= 2 cos (θ) \frac{1}{sin ( θ )} \cdot \frac{sin ( θ )}{cos ( θ )}

2 cos (θ) \frac{1}{sin ( θ )} \cdot \frac{sin ( θ )}{cos ( θ )} = 2

2 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 ) \cdot 2 cos ( θ )}{sin ( θ ) cos ( θ )}

Cancel the common factor:

sin (θ)

= \frac{1 \cdot 2 cos ( θ )}{cos ( θ )}

Cancel the common factor:

cos (θ)

= 1 \cdot 2

Multiply the numbers:

1 \cdot 2 = 2

= 2

= 2

= 2

We showed that the two sides could take the same form

\Rightarrow True