Question #696f9

1 Answer
Nghi N.
Jun 19, 2016

Prove trig identity

Explanation:

Use these trig identities:
#1. tan (a + b) = (tan a + tan b)/(1 - tan a.tan b)#
#2. 1 + cos 2a = 2cos^2 a#
3. sin 2a = 2sin a.cos a
Apply the first trig identity to the left side of the equation:
#LS = tan (45 + tan (t/2)) = (1 + tan (t/2))/(1 - tan (t/2)) = #
Replace #tan (t/2)# by #(sin t/2)/(cos t/2)#, we get:
#LS = (cos (t/2) + sin (t/2))/(cos (t/2) - sin (t/2)= # (1)
Now, transform the right side of the equation:
#RS = (1 + cos t + sin t)/( 1 + cos t - sin t) =#
Replace
#1 + cos t# by #2cos^2 (t/2)#
#sin t = 2sin (t/2).cos (t/2)#, we get
#RS = (2cos^2 (t/2)(cos (t/2) + sin (t/2)))/(2cos^2 (t/2)(cos (t/2) - sin (t/2)#.
#RS = (cos (t/2) + sin (t/2))/(cos (t/2) - sin (t/2)) = LS#
The trig identity is proven.

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