Is (tan(θ)+6)/(sec(θ))=6cos(θ)+sin(θ) ?

The answer to whether (tan(θ)+6)/(sec(θ))=6cos(θ)+sin(θ) is True

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prove (tan(θ)+6)/(sec(θ))=6cos(θ)+sin(θ)

prove

\frac{tan ( θ ) + 6}{sec ( θ )} = 6 cos (θ) + sin (θ)

True

Solution steps

\frac{tan ( θ ) + 6}{sec ( θ )} = 6 cos (θ) + sin (θ)

Manipulating left side

\frac{tan ( θ ) + 6}{sec ( θ )}

Express with sin, cos

\frac{6 + tan ( θ )}{sec ( θ )}

Use the basic trigonometric identity:

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

= \frac{6 + \frac{s i n ( θ )}{c o s ( θ )}}{sec ( θ )}

Use the basic trigonometric identity:

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

= \frac{6 + \frac{s i n ( θ )}{c o s ( θ )}}{\frac{1}{c o s ( θ )}}

Simplify

\frac{6 + \frac{s i n ( θ )}{c o s ( θ )}}{\frac{1}{c o s ( θ )}} : 6 cos (θ) + sin (θ)

\frac{6 + \frac{s i n ( θ )}{c o s ( θ )}}{\frac{1}{c o s ( θ )}}

Apply the fraction rule:

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

= \frac{( 6 + \frac{s i n ( θ )}{c o s ( θ )} ) cos ( θ )}{1}

Join

6 + \frac{sin ( θ )}{cos ( θ )} : \frac{6 cos ( θ ) + sin ( θ )}{cos ( θ )}

6 + \frac{sin ( θ )}{cos ( θ )}

Convert element to fraction:

6 = \frac{6 cos ( θ )}{cos ( θ )}

= \frac{6 cos ( θ )}{cos ( θ )} + \frac{sin ( θ )}{cos ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{6 cos ( θ ) + sin ( θ )}{cos ( θ )}

= \frac{\frac{6 c o s ( θ ) + s i n ( θ )}{c o s ( θ )} cos ( θ )}{1}

Apply the fraction rule:

\frac{a}{1} = a

= \frac{6 cos ( θ ) + sin ( θ )}{cos ( θ )} cos (θ)

Multiply fractions:

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

= \frac{( 6 cos ( θ ) + sin ( θ ) ) cos ( θ )}{cos ( θ )}

Cancel the common factor:

cos (θ)

= 6 cos (θ) + sin (θ)

= 6 cos (θ) + sin (θ)

= sin (θ) + 6 cos (θ)

= 6 cos (θ) + sin (θ)

We showed that the two sides could take the same form

\Rightarrow True

sin (x) + 5 = 4

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

sin^{2} (x) - 1 = 2 tan (x) - 1

2 - csc^{2} (θ) = 1 + cot^{2} (θ)

sin (\frac{π}{6} + b) + cos (\frac{π}{3} + b) = cos (b)

Is (tan(θ)+6)/(sec(θ))=6cos(θ)+sin(θ) ?
The answer to whether (tan(θ)+6)/(sec(θ))=6cos(θ)+sin(θ) is True