Is cot(x)sec(x)(1-cos^2(x))=sin(x) ?

The answer to whether cot(x)sec(x)(1-cos^2(x))=sin(x) is True

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prove cot(x)sec(x)(1-cos^2(x))=sin(x)

prove

cot (x) sec (x) (1 - cos^{2} (x)) = sin (x)

True

Solution steps

cot (x) sec (x) (1 - cos^{2} (x)) = sin (x)

Manipulating left side

cot (x) sec (x) (1 - cos^{2} (x))

Express with sin, cos

(1 - cos^{2} (x)) cot (x) sec (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

(1 - cos^{2} (x)) \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} : \frac{1 - cos ^{2} ( x )}{sin ( x )}

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

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

1 \cdot (1 - cos^{2} (x)) = 1 - cos^{2} (x)

1 \cdot (1 - cos^{2} (x))

Multiply:

1 \cdot (1 - cos^{2} (x)) = (1 - cos^{2} (x))

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

Remove parentheses:

(a) = a

= 1 - cos^{2} (x)

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

1 = cos^{2} (x) + sin^{2} (x)

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

= \frac{sin ^{2} ( x )}{sin ( x )}

Cancel the common factor:

sin (x)

= sin (x)

= sin (x)

We showed that the two sides could take the same form

\Rightarrow True

tan (x) + 2 = 2 + sec (x) sin (x)

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

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

\frac{1 - sin ^{2} ( θ )}{sin ^{2} ( θ )} = cot^{2} (θ)

sin (x) = tan (x) cos (x)

Is cot(x)sec(x)(1-cos^2(x))=sin(x) ?
The answer to whether cot(x)sec(x)(1-cos^2(x))=sin(x) is True