\operatorname { sec } ( \frac { \pi } { 4 } + \theta ) \operatorname {

1.	\operatorname { sec } ( \frac { \pi } { 4 } + \theta ) \operatorname { sec } ( \frac { \pi } { 4 } - \theta ) = 2 \operatorname { sec } 2 \theta
Answer» sec(π/4 + A) sec(π/4 - A) = 1 / [ cos(π/4 + A) cos(π/4 - A)] = 1/ [ {cos(π/4)cos A - sin(π/4) sin A}{cos(π/4)cos A + sin(π/4) sin A}] = 1/ [ (1/√2)cos A - sin A}(1/√2){cos A + sin A}] = 2 / [ (cos A - sin A)(cos A + sin A)] = 2 / [ cos^2(A) - sin^2(A)] = 2 / cos(2A) = 2 sec(2A)

\operatorname { sec } ( \frac { \pi } { 4 } + \theta ) \operatorname { sec } ( \frac { \pi } { 4 } - \theta ) = 2 \operatorname { sec } 2 \theta

Answer»

sec(π/4 + A) sec(π/4 - A)

= 1 / [ cos(π/4 + A) cos(π/4 - A)]

= 1/ [ {cos(π/4)cos A - sin(π/4) sin A}{cos(π/4)cos A + sin(π/4) sin A}]

= 1/ [ (1/√2)cos A - sin A}(1/√2){cos A + sin A}]

= 2 / [ (cos A - sin A)(cos A + sin A)]

= 2 / [ cos^2(A) - sin^2(A)]

= 2 / cos(2A)

= 2 sec(2A)