Expand the Trigonometric Expression (r-t+n)^2

${(r - t + n)}^{2}$

Step 1

Rewrite ${(r - t + n)}^{2}$ as $(r - t + n) (r - t + n)$ .

$(r - t + n) (r - t + n)$

Step 2

Expand $(r - t + n) (r - t + n)$ by multiplying each term in the first expression by each term in the second expression.

$r \cdot r + r (- t) + r n - t r - t (- t) - t n + n r + n (- t) + n \cdot n$

Step 3

Simplify each term.

Multiply $r$ by $r$ .

$r^{2} + r (- t) + r n - t r - t (- t) - t n + n r + n (- t) + n \cdot n$

Rewrite using the commutative property of multiplication.

$r^{2} - r t + r n - t r - t (- t) - t n + n r + n (- t) + n \cdot n$

Rewrite using the commutative property of multiplication.

$r^{2} - r t + r n - t r - 1 \cdot - 1 t \cdot t - t n + n r + n (- t) + n \cdot n$

Multiply $t$ by $t$ by adding the exponents.

Move $t$ .

$r^{2} - r t + r n - t r - 1 \cdot - 1 (t \cdot t) - t n + n r + n (- t) + n \cdot n$

Multiply $t$ by $t$ .

$r^{2} - r t + r n - t r - 1 \cdot - 1 t^{2} - t n + n r + n (- t) + n \cdot n$

Multiply $- 1$ by $- 1$ .

$r^{2} - r t + r n - t r + 1 t^{2} - t n + n r + n (- t) + n \cdot n$

Multiply $t^{2}$ by $1$ .

$r^{2} - r t + r n - t r + t^{2} - t n + n r + n (- t) + n \cdot n$

Rewrite using the commutative property of multiplication.

$r^{2} - r t + r n - t r + t^{2} - t n + n r - n t + n \cdot n$

Multiply $n$ by $n$ .

$r^{2} - r t + r n - t r + t^{2} - t n + n r - n t + n^{2}$

Step 4

Subtract $t r$ from $- r t$ .

Move $t$ .

$r^{2} - r t - 1 r t + r n + t^{2} - t n + n r - n t + n^{2}$

Subtract $r t$ from $- r t$ .

$r^{2} - 2 r t + r n + t^{2} - t n + n r - n t + n^{2}$

Step 5

Add $r n$ and $n r$ .

Reorder $r$ and $n$ .

$r^{2} - 2 r t + t^{2} - t n + n r + n r - n t + n^{2}$

Add $n r$ and $n r$ .

$r^{2} - 2 r t + t^{2} - t n + 2 n r - n t + n^{2}$

Step 6

Subtract $n t$ from $- t n$ .

Move $t$ .

$r^{2} - 2 r t + t^{2} - 1 n t - n t + 2 n r + n^{2}$

Subtract $n t$ from $- 1 n t$ .

$r^{2} - 2 r t + t^{2} - 2 n t + 2 n r + n^{2}$

Expand the Trigonometric Expression (r-t+n)^2

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