Question

tan (π/4 + X/2 ) = tan X + sec X
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Answer 1

Answer:

RHS = tan (pi/4 + x/2)

=tan (45 + x/2)

= (tan 45 + tan x/2)/(1 - tan 45*tan x/2)

= (1 + tan x/2)/(1 - tan x/2)

= [1 + sin (x/2)/cos (x/2)]/[1 - sin (x/2)/cos (x/2)]

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

= [cos (x/2) + sin (x/2)]/[cos (x/2) - sin (x/2)] … (1)

LHS = sec x + tan x

= (1/cos x) + (sin x/cos x)

= (1 + sin x)/cos x

= [(sin^2 (x/2) + cos^2 (x/2) + 2 sin (x/2).cos (x/2)]/[cos^2 (x/2) - sin^2 (x/2)]

= [(sin (x/2) + cos (x/2)]^2/[cos^2 (x/2) - sin^2 (x/2)]

= [cos (x/2) + sin (x/2)]^2/[cos (x/2) - sin (x/2)]*[cos (x/2) + sin (x/2)]

= [cos (x/2) + sin (x/2)]/[cos (x/2) - sin (x/2)] … (2)

Both (1) and (2) are the same, hence it is proved that sec x + tan x = tan (pi/4 + x/2).