Is 25*sec^2(5x)=25+(5tan(5x))^2 ?

The answer to whether 25*sec^2(5x)=25+(5tan(5x))^2 is True

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prove 25*sec^2(5x)=25+(5tan(5x))^2

prove

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

True

Solution steps

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

Manipulating right side

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

Simplify

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

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

Apply exponent rule:

(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)

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Multiply the numbers:

25 \cdot 1 = 25

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

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

Simplify

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

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

Group like terms

= 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)

We showed that the two sides could take the same form

\Rightarrow True

Is 25*sec^2(5x)=25+(5tan(5x))^2 ?
The answer to whether 25*sec^2(5x)=25+(5tan(5x))^2 is True