Is (cos(x)sin(x))/(tan(x))=1-sin^2(x) ?

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

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prove (cos(x)sin(x))/(tan(x))=1-sin^2(x)

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

= \frac{cos ( x ) sin ( x )}{\frac{s i n ( x )}{c o s ( x )}}

Simplify

\frac{cos ( x ) sin ( x )}{\frac{s i n ( x )}{c o s ( x )}} : cos^{2} (x)

\frac{cos ( x ) sin ( x )}{\frac{s i n ( x )}{c o s ( x )}}

Apply the fraction rule:

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

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

Cancel the common factor:

sin (x)

= cos (x) cos (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

= cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cos^{2} (x)

= cos^{2} (x)

= cos^{2} (x)

Rewrite using trig identities

cos^{2} (x)

Use the Pythagorean identity:

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

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

= 1 - sin^{2} (x)

= 1 - sin^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

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

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

(3 sin (θ) + 3 cos (θ))^{2} = 9 + 9 sin (2 θ)

tan (x) - \frac{sec ^{2} ( x )}{tan ( x )} = - cot (x)

\frac{tan ^{2} ( θ ) + 1}{tan ^{2} ( θ )} = csc^{2} (θ)

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