What is sqrt((5sec(x))^2-5^2) ?

The solution to sqrt((5sec(x))^2-5^2) is 5tan(x)

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simplify sqrt((5sec(x))^2-5^2)

simplify

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

5 tan (x)

Solution steps

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

Apply exponent rule:

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

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

Factor

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

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

Apply radical rule:

assuming

a \geq 0, b \geq 0

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

25 = 5

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

Use the Pythagorean identity:

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

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

= 5 tan^{2} (x)

tan^{2} (x) = tan (x)

= 5 tan (x)

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What is sqrt((5sec(x))^2-5^2) ?
The solution to sqrt((5sec(x))^2-5^2) is 5tan(x)