Is 1/(sec(x)-tan(x))=sec(x)+tan(x) ?

The answer to whether 1/(sec(x)-tan(x))=sec(x)+tan(x) is True

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prove 1/(sec(x)-tan(x))=sec(x)+tan(x)

prove

\frac{1}{sec ( x ) - tan ( x )} = sec (x) + tan (x)

True

Solution steps

\frac{1}{sec ( x ) - tan ( x )} = sec (x) + tan (x)

Manipulating left side

\frac{1}{sec ( x ) - tan ( x )}

Rewrite using trig identities

\frac{1}{sec ( x ) - tan ( x )}

Use the Pythagorean identity:

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

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

= \frac{sec ^{2} ( x ) - tan ^{2} ( x )}{sec ( x ) - tan ( x )}

Simplify

\frac{sec ^{2} ( x ) - tan ^{2} ( x )}{sec ( x ) - tan ( x )} : sec (x) + tan (x)

\frac{sec ^{2} ( x ) - tan ^{2} ( x )}{sec ( x ) - tan ( x )}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

= \frac{( sec ( x ) + tan ( x ) ) ( sec ( x ) - tan ( x ) )}{sec ( x ) - tan ( x )}

Cancel the common factor:

sec (x) - tan (x)

= sec (x) + tan (x)

= sec (x) + tan (x)

= sec (x) + tan (x)

We showed that the two sides could take the same form

\Rightarrow True

cos (9 0^{\circ})

cos (- \frac{3 π}{4})

sin (- \frac{5 π}{6})

sin (3 7^{\circ})

sin (2 x) = 1

Is 1/(sec(x)-tan(x))=sec(x)+tan(x) ?
The answer to whether 1/(sec(x)-tan(x))=sec(x)+tan(x) is True