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

sec (x) - tan (x) sin (x) = cos (x)

sin (θ) cot (θ) = cos (θ)

tan (θ) csc (θ) cos (θ) = 1

1 + tan^{2} (θ) = sec^{2} (θ)

\frac{sin ( π - x )}{sin ( x + \frac{π}{2} )} = tan (x)

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