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

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Manipular el lado izquierdo

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

Expandir

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

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

Aplicar la formula del binomio al cuadrado:

(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)

Simplificar

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)

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

Multiplicar los numeros:

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)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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)