Is (cot(x)+1)^2-2cot(x)=csc^2(x) ?

The answer to whether (cot(x)+1)^2-2cot(x)=csc^2(x) is True

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prove (cot(x)+1)^2-2cot(x)=csc^2(x)

prove

(cot (x) + 1)^{2} - 2 cot (x) = csc^{2} (x)

True

Solution steps

(cot (x) + 1)^{2} - 2 cot (x) = csc^{2} (x)

Manipulating left side

(cot (x) + 1)^{2} - 2 cot (x)

Simplify

(1 + cot (x))^{2} - 2 cot (x) : cot^{2} (x) + 1

(1 + cot (x))^{2} - 2 cot (x)

(1 + cot (x))^{2} : 1 + 2 cot (x) + cot^{2} (x)

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = 1, b = cot (x)

= 1^{2} + 2 \cdot 1 \cdot cot (x) + cot^{2} (x)

Simplify

1^{2} + 2 \cdot 1 \cdot cot (x) + cot^{2} (x) : 1 + 2 cot (x) + cot^{2} (x)

1^{2} + 2 \cdot 1 \cdot cot (x) + cot^{2} (x)

Apply rule

1^{a} = 1

1^{2} = 1

= 1 + 2 \cdot 1 \cdot cot (x) + cot^{2} (x)

Multiply the numbers:

2 \cdot 1 = 2

= 1 + 2 cot (x) + cot^{2} (x)

= 1 + 2 cot (x) + cot^{2} (x)

= 1 + 2 cot (x) + cot^{2} (x) - 2 cot (x)

Simplify

1 + 2 cot (x) + cot^{2} (x) - 2 cot (x) : cot^{2} (x) + 1

1 + 2 cot (x) + cot^{2} (x) - 2 cot (x)

Group like terms

= 2 cot (x) + cot^{2} (x) - 2 cot (x) + 1

Add similar elements:

2 cot (x) - 2 cot (x) = 0

= cot^{2} (x) + 1

= cot^{2} (x) + 1

= 1 + cot^{2} (x)

Rewrite using trig identities

1 + cot^{2} (x)

Use the Pythagorean identity:

1 + cot^{2} (x) = csc^{2} (x)

= csc^{2} (x)

= csc^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

p ro v e cos (θ) = \frac{1}{3}

p ro v e sec (θ) - \frac{tan ( θ )}{csc ( θ )} = cos (θ)

p ro v e sec^{2} (x) = (1 + tan^{2} (x))

p ro v e cot^{2} (x) sec^{2} (x) = cot^{2} (x) + 1

p ro v e \frac{sin ( x )}{cos ( x ) + sin ( x )} = 1

Is (cot(x)+1)^2-2cot(x)=csc^2(x) ?
The answer to whether (cot(x)+1)^2-2cot(x)=csc^2(x) is True