Is 1-sec(x)cos^3(x)=sin^2(x) ?

The answer to whether 1-sec(x)cos^3(x)=sin^2(x) is True

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prove 1-sec(x)cos^3(x)=sin^2(x)

prove

1 - sec (x) cos^{3} (x) = sin^{2} (x)

True

Solution steps

1 - sec (x) cos^{3} (x) = sin^{2} (x)

Manipulating left side

1 - sec (x) cos^{3} (x)

Express with sin, cos

1 - cos^{3} (x) sec (x)

Use the basic trigonometric identity:

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

= 1 - cos^{3} (x) \frac{1}{cos ( x )}

Simplify

1 - cos^{3} (x) \frac{1}{cos ( x )} : 1 - cos^{2} (x)

1 - cos^{3} (x) \frac{1}{cos ( x )}

cos^{3} (x) \frac{1}{cos ( x )} = cos^{2} (x)

cos^{3} (x) \frac{1}{cos ( x )}

Multiply fractions:

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

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

Multiply:

1 \cdot cos^{3} (x) = cos^{3} (x)

= \frac{cos ^{3} ( x )}{cos ( x )}

Cancel the common factor:

cos (x)

= cos^{2} (x)

= 1 - cos^{2} (x)

= 1 - cos^{2} (x)

= 1 - cos^{2} (x)

Rewrite using trig identities

1 - cos^{2} (x)

Use the Pythagorean identity:

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

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

= sin^{2} (x)

= sin^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

Is 1-sec(x)cos^3(x)=sin^2(x) ?
The answer to whether 1-sec(x)cos^3(x)=sin^2(x) is True