Is tan(6x)sec(3x)=2sin(3x)sec(6x) ?

The answer to whether tan(6x)sec(3x)=2sin(3x)sec(6x) is True

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prove tan(6x)sec(3x)=2sin(3x)sec(6x)

prove

tan (6 x) sec (3 x) = 2 sin (3 x) sec (6 x)

True

Solution steps

tan (6 x) sec (3 x) = 2 sin (3 x) sec (6 x)

Manipulating left side

tan (6 x) sec (3 x)

Express with sin, cos

sec (3 x) tan (6 x)

Use the basic trigonometric identity:

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

= \frac{1}{cos ( 3 x )} tan (6 x)

Use the basic trigonometric identity:

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

= \frac{1}{cos ( 3 x )} \cdot \frac{sin ( 6 x )}{cos ( 6 x )}

Simplify

\frac{1}{cos ( 3 x )} \cdot \frac{sin ( 6 x )}{cos ( 6 x )} : \frac{sin ( 6 x )}{cos ( 3 x ) cos ( 6 x )}

\frac{1}{cos ( 3 x )} \cdot \frac{sin ( 6 x )}{cos ( 6 x )}

Multiply fractions:

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

= \frac{1 \cdot sin ( 6 x )}{cos ( 3 x ) cos ( 6 x )}

Multiply:

1 \cdot sin (6 x) = sin (6 x)

= \frac{sin ( 6 x )}{cos ( 3 x ) cos ( 6 x )}

= \frac{sin ( 6 x )}{cos ( 3 x ) cos ( 6 x )}

= \frac{sin ( 6 x )}{cos ( 3 x ) cos ( 6 x )}

Rewrite using trig identities

\frac{sin ( 6 x )}{cos ( 3 x ) cos ( 6 x )}

= \frac{sin ( 2 \cdot 3 x )}{cos ( 3 x ) cos ( 6 x )}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= \frac{2 sin ( 3 x ) cos ( 3 x )}{cos ( 3 x ) cos ( 6 x )}

Cancel the common factor:

cos (3 x)

= \frac{2 sin ( 3 x )}{cos ( 6 x )}

= \frac{2 sin ( 3 x )}{cos ( 6 x )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{2 sin ( 3 x )}{\frac{1}{s e c ( 6 x )}}

Simplify

\frac{2 sin ( 3 x )}{\frac{1}{s e c ( 6 x )}}

Apply the fraction rule:

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

= \frac{2 sin ( 3 x ) sec ( 6 x )}{1}

Apply rule

\frac{a}{1} = a

= 2 sin (3 x) sec (6 x)

2 sin (3 x) sec (6 x)

2 sin (3 x) sec (6 x)

We showed that the two sides could take the same form

\Rightarrow True

sec (θ) (cos (θ) + cot (θ)) = csc (θ)

cos (θ) = \frac{11}{6}

1 - sin^{2} (\frac{x}{4}) = cos (\frac{x}{2})

\frac{csc ( θ ) + sec ( θ )}{csc ( θ )} = csc (θ)

sec (x) + 1 = \frac{cos ( x ) + 1}{cos ( x )}

Is tan(6x)sec(3x)=2sin(3x)sec(6x) ?
The answer to whether tan(6x)sec(3x)=2sin(3x)sec(6x) is True