Is sec(pi/4+a)sec(pi/4-a)=2sec(2a) ?

The answer to whether sec(pi/4+a)sec(pi/4-a)=2sec(2a) is True

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prove sec(pi/4+a)sec(pi/4-a)=2sec(2a)

prove

sec (\frac{π}{4} + a) sec (\frac{π}{4} - a) = 2 sec (2 a)

True

Solution steps

sec (\frac{π}{4} + a) sec (\frac{π}{4} - a) = 2 sec (2 a)

Manipulating left side

sec (\frac{π}{4} + a) sec (\frac{π}{4} - a)

Rewrite using trig identities

= sec (\frac{π}{4} + a) \frac{2}{cos ( a ) + sin ( a )}

Rewrite using trig identities

= \frac{2}{cos ( a ) - sin ( a )} \cdot \frac{2}{cos ( a ) + sin ( a )}

Simplify

\frac{2}{cos ( a ) - sin ( a )} \cdot \frac{2}{cos ( a ) + sin ( a )} : \frac{2}{( cos ( a ) - sin ( a ) ) ( cos ( a ) + sin ( a ) )}

= \frac{2}{( cos ( a ) - sin ( a ) ) ( cos ( a ) + sin ( a ) )}

Manipulating right side

2 sec (2 a)

Express with sin, cos

= \frac{2}{cos ( 2 a )}

Rewrite using trig identities

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

Factor

cos^{2} (a) - sin^{2} (a) : (cos (a) + sin (a)) (cos (a) - sin (a))

= \frac{2}{( cos ( a ) + sin ( a ) ) ( cos ( a ) - sin ( a ) )}

We showed that the two sides could take the same form

\Rightarrow True

sin^{4} (x) = sec^{2} (x) tan^{2} (x) + sec^{2} (x)

(1 + cos (x)) (1 - sec (x)) = - 1

cot^{2} (\frac{v}{2}) = \frac{( sec ( v ) + 1 )}{( sec ( v ) - 1 )}

csc (θ) (1 - cos (θ)) (1 + cos (θ)) = sin (θ)

sec (θ) - \frac{1}{sec ( θ )} = sin (θ) tan (θ)

Is sec(pi/4+a)sec(pi/4-a)=2sec(2a) ?
The answer to whether sec(pi/4+a)sec(pi/4-a)=2sec(2a) is True