Simplify cos(t)(sec(t)-cos(t))

$\cos (t) (\sec (t) - \cos (t))$

Rewrite $\sec (t)$ in terms of sines and cosines.

$\cos (t) (\frac{1}{\cos (t)} - \cos (t))$

Apply the distributive property.

$\cos (t) \frac{1}{\cos (t)} + \cos (t) (- \cos (t))$

Cancel the common factor of $\cos (t)$ .

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Cancel the common factor.

$\cos (t) \frac{1}{\cos (t)} + \cos (t) (- \cos (t))$

Rewrite the expression.

$1 + \cos (t) (- \cos (t))$

Rewrite using the commutative property of multiplication.

$1 - \cos (t) \cos (t)$

Multiply $- \cos (t) \cos (t)$ .

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Raise $\cos (t)$ to the power of $1$ .

$1 - (\cos^{1} (t) \cos (t))$

Raise $\cos (t)$ to the power of $1$ .

$1 - (\cos^{1} (t) \cos^{1} (t))$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 - {\cos (t)}^{1 + 1}$

Add $1$ and $1$ .

$1 - \cos^{2} (t)$

Apply pythagorean identity.

$\sin^{2} (t)$

Simplify cos(t)(sec(t)-cos(t))

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