Is sin(x)+cos(x)=cos(x)(1+tan(x)) ?

The answer to whether sin(x)+cos(x)=cos(x)(1+tan(x)) is True

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

prove

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

True

Solution steps

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

Manipulating right side

cos (x) (1 + tan (x))

Express with sin, cos

(1 + tan (x)) cos (x)

Use the basic trigonometric identity:

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

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

Simplify

(1 + \frac{sin ( x )}{cos ( x )}) cos (x) : cos (x) + sin (x)

(1 + \frac{sin ( x )}{cos ( x )}) cos (x)

Join

1 + \frac{sin ( x )}{cos ( x )} : \frac{cos ( x ) + sin ( x )}{cos ( x )}

1 + \frac{sin ( x )}{cos ( x )}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot cos (x) = cos (x)

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= cos (x) + sin (x)

= cos (x) + sin (x)

= cos (x) + sin (x)

= sin (x) + cos (x)

We showed that the two sides could take the same form

\Rightarrow True

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

cot (θ) sec^{2} (θ) - cot (θ) = tan (θ)

2 cos (x - \frac{π}{4}) = cos (x) + sin (x)

cos^{4} (4 x) - sin^{4} (4 x) = cos (8 x)

(cos^{2} (x) - 1) (tan^{2} (x) + 1) = - tan^{2} (x)

Is sin(x)+cos(x)=cos(x)(1+tan(x)) ?
The answer to whether sin(x)+cos(x)=cos(x)(1+tan(x)) is True