What is sec^2(5x-In(1/x)) ?

The solution to sec^2(5x-In(1/x)) is sec^2((nI-5x^2)/x)

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simplify sec^2(5x-In(1/x))

simplify

sec^{2} (5 x - I n (\frac{1}{x}))

sec^{2} (\frac{n I - 5 x ^{2}}{x})

Solution steps

sec^{2} (5 x - I n (\frac{1}{x}))

Rewrite using trig identities

= (\frac{1}{cos ( \frac{n I}{x} ) cos ( 5 x ) + sin ( \frac{n I}{x} ) sin ( 5 x )})^{2}

Simplify

(\frac{1}{cos ( \frac{n I}{x} ) cos ( 5 x ) + sin ( \frac{n I}{x} ) sin ( 5 x )})^{2} : \frac{1}{( cos ( \frac{n I}{x} ) cos ( 5 x ) + sin ( \frac{n I}{x} ) sin ( 5 x ) ) ^{2}}

= \frac{1}{( cos ( \frac{n I}{x} ) cos ( 5 x ) + sin ( \frac{n I}{x} ) sin ( 5 x ) ) ^{2}}

Use the Angle Difference identity:

cos (s) cos (t) + sin (s) sin (t) = cos (s - t)

= \frac{1}{cos ^{2} ( \frac{n I}{x} - 5 x )}

cos^{2} (\frac{n I}{x} - 5 x) = cos^{2} (\frac{n I - 5 x ^{2}}{x})

= \frac{1}{cos ^{2} ( \frac{n I - 5 x ^{2}}{x} )}

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

= sec^{2} (\frac{n I - 5 x ^{2}}{x})

sin (\frac{π}{2} + \frac{x}{2})

e^{x^{2}} + \frac{1}{sin ( x )} d x

sin (\frac{sin ( x )}{cos ( x )})

cos (2 π + θ)

3 (cos (- 3 π) - x sin (- 3 π))

What is sec^2(5x-In(1/x)) ?
The solution to sec^2(5x-In(1/x)) is sec^2((nI-5x^2)/x)