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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Expand

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

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

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

Manipulating right side

csc^{2} (x) - 2 cot (x)

Rewrite using trig identities

csc^{2} (x) - 2 cot (x)

Use the Pythagorean identity:

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

p ro v e \frac{1}{cot ( x ) ( 1 - cos ( 2 x ) )} = csc (2 x)

p ro v e (tan (θ) + cot (θ)) cos (θ) = csc (θ)

p ro v e cos^{4} (t) - sin^{4} (t) = 1 - 2 sin^{2} (t)

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

p ro v e \frac{tan ( x ) + tan ( y )}{cot ( x ) + cot ( y )} = tan (x) tan (y)

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