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

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

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

prove

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

True

Solution steps

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

Manipulating right side

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

tan (x) + cot (x) = 1

tan (a) = \frac{cos ( a )}{sin ( a ) cot ^{2} ( a )}

sin (θ - π) = - sin (θ)

csc (u) sin (u) - sin^{2} (u) = cos^{2} (u)

\frac{sin ( x ) cos ( x )}{1 - cos ^{2} ( x )} = cot (x)

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