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verificar 25sec^2(5x)=25+(5tan(5x))^2

Solución

verificar

25 sec^{2} (5 x) = 25 + (5 tan (5 x))^{2}

Verdadero

Pasos de solución

25 sec^{2} (5 x) = 25 + (5 tan (5 x))^{2}

Manipular el lado izquierdo

25 + (5 tan (5 x))^{2}

Simplificar

(5 tan (5 x))^{2} + 25 : 25 tan^{2} (5 x) + 25

(5 tan (5 x))^{2} + 25

Aplicar las leyes de los exponentes:

(a \cdot b)^{n} = a^{n} b^{n}

= 5^{2} tan^{2} (5 x) + 25

5^{2} = 25

= 25 tan^{2} (5 x) + 25

= 25 + 25 tan^{2} (5 x)

Re-escribir usando identidades trigonométricas

25 + 25 tan^{2} (5 x)

Utilizar la identidad pitagórica:

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

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

= 25 + 25 (sec^{2} (5 x) - 1)

Simplificar

25 + 25 (sec^{2} (5 x) - 1) : 25 sec^{2} (5 x)

25 + 25 (sec^{2} (5 x) - 1)

Expandir

25 (sec^{2} (5 x) - 1) : 25 sec^{2} (5 x) - 25

25 (sec^{2} (5 x) - 1)

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = 25, b = sec^{2} (5 x), c = 1

= 25 sec^{2} (5 x) - 25 \cdot 1

Multiplicar los numeros:

25 \cdot 1 = 25

= 25 sec^{2} (5 x) - 25

= 25 + 25 sec^{2} (5 x) - 25

Simplificar

25 + 25 sec^{2} (5 x) - 25 : 25 sec^{2} (5 x)

25 + 25 sec^{2} (5 x) - 25

Agrupar términos semejantes

= 25 sec^{2} (5 x) + 25 - 25

25 - 25 = 0

= 25 sec^{2} (5 x)

= 25 sec^{2} (5 x)

= 25 sec^{2} (5 x)

= 25 sec^{2} (5 x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

tan (x) = csc (2 x) - cot (2 x)

sin^{2} (- x) + cos^{2} (- x) = 1

\frac{sec ( x ) - cos ( x )}{tan ( x )} = sin (x)

sin (2 x) = 2 cos (x) sin (x)

cos (x + π) = - cos (x)