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

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

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

prove

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

True

Solution steps

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

Manipulating left 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)

Manipulating right 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)

We showed that the two sides could take the same form

\Rightarrow True

csc (t) cot (t) = \frac{1 + cot ^{2} ( t )}{sec ( t )}

csc (a) sec (a) = 2 csc (2 a)

tan^{2} (\frac{x}{2}) = \frac{sec ( x ) - 1}{sec ( x ) + 1}

tan^{2} (2 x) + 1 = sec^{2} (2 x)

sin (4 θ) = 4 sin (θ) cos (θ) cos (2 θ)

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