Trigonometric

Here is a step by step explanation! But, prepare yourself, it is very lengthy!!:
Verify the following identity:
(1+tan(3 t))/(1-tan(3 t))  =  tan(pi/4+3 t)

 

Multiply both sides by 1-tan(3 t):
1+tan(3 t)  =  ^?tan(pi/4+3 t) (1-tan(3 t))

 

Write tangent as sine/cosine:
1+(sin(3 t))/(cos(3 t))  =  ^?(1-(sin(3 t))/(cos(3 t))) (sin(pi/4+3 t))/(cos(pi/4+3 t))

 

(1-((sin(3 t))/(cos(3 t)))) ((sin(pi/4+3 t))/(cos(pi/4+3 t))) = ((1-(sin(3 t))/(cos(3 t))) sin(pi/4+3 t))/(cos(pi/4+3 t)):
1+(sin(3 t))/(cos(3 t))  =  ^?(sin(pi/4+3 t) (1-(sin(3 t))/(cos(3 t))))/(cos(pi/4+3 t))

 

Put 1+(sin(3 t))/(cos(3 t)) over the common denominator cos(3 t): 1+(sin(3 t))/(cos(3 t))  =  (cos(3 t)+sin(3 t))/(cos(3 t)):
(cos(3 t)+sin(3 t))/(cos(3 t))  =  ^?(sin(pi/4+3 t) (1-(sin(3 t))/(cos(3 t))))/(cos(pi/4+3 t))

 

Put 1-(sin(3 t))/(cos(3 t)) over the common denominator cos(3 t): 1-(sin(3 t))/(cos(3 t))  =  (cos(3 t)-sin(3 t))/(cos(3 t)):
(cos(3 t)+sin(3 t))/(cos(3 t))  =  ^?((cos(3 t)-sin(3 t))/(cos(3 t)) sin(pi/4+3 t))/(cos(pi/4+3 t))

 

Cross multiply:
cos(3 t) cos(pi/4+3 t) (cos(3 t)+sin(3 t))  =  ^?cos(3 t) sin(pi/4+3 t) (cos(3 t)-sin(3 t))

 

Divide both sides by cos(3 t):
cos(pi/4+3 t) (cos(3 t)+sin(3 t))  =  ^?sin(pi/4+3 t) (cos(3 t)-sin(3 t))

 

cos(pi/4+3 t) = cos(pi/4) cos(3 t)-sin(pi/4) sin(3 t):
cos(pi/4) cos(3 t)-sin(pi/4) sin(3 t) (cos(3 t)+sin(3 t))  =  ^?sin(pi/4+3 t) (cos(3 t)-sin(3 t))

 

cos(pi/4) = 1/sqrt(2):
(1/sqrt(2) cos(3 t)-sin(pi/4) sin(3 t)) (cos(3 t)+sin(3 t))  =  ^?sin(pi/4+3 t) (cos(3 t)-sin(3 t))

 

sin(pi/4) = 1/sqrt(2):
((1 cos(3 t))/sqrt(2)-1/sqrt(2) sin(3 t)) (cos(3 t)+sin(3 t))  =  ^?sin(pi/4+3 t) (cos(3 t)-sin(3 t))

 

((1 cos(3 t))/sqrt(2)-(1 sin(3 t))/sqrt(2)) (cos(3 t)+sin(3 t)) = cos(3 t)^2/sqrt(2)-sin(3 t)^2/sqrt(2):
cos(3 t)^2/sqrt(2)-sin(3 t)^2/sqrt(2)  =  ^?sin(pi/4+3 t) (cos(3 t)-sin(3 t))

 

cos(3 t)^2 = 1/2 (1+cos(6 t)):
1/sqrt(2) (1+cos(6 t))/2-sin(3 t)^2/sqrt(2)  =  ^?sin(pi/4+3 t) (cos(3 t)-sin(3 t))

 

(1+cos(6 t))/2 = 1/2+1/2 cos(6 t):
1/sqrt(2) 1/2+(cos(6 t))/2-sin(3 t)^2/sqrt(2)  =  ^?sin(pi/4+3 t) (cos(3 t)-sin(3 t))

 

sin(3 t)^2 = 1/2 (1-cos(6 t)):
(1 (1/2+(cos(6 t))/2))/(sqrt(2))-1/sqrt(2) (1-cos(6 t))/2  =  ^?sin(pi/4+3 t) (cos(3 t)-sin(3 t))

 

(1-cos(6 t))/2 = 1/2-1/2 cos(6 t):
(1 (1/2+(cos(6 t))/2))/(sqrt(2))-1/sqrt(2) 1/2-(cos(6 t))/(2)  =  ^?sin(pi/4+3 t) (cos(3 t)-sin(3 t))

 

(1 (1/2+(cos(6 t))/2))/(sqrt(2)) = 1/(2 sqrt(2))+(cos(6 t))/(2 sqrt(2)):
(1)/(2 sqrt(2))+(1)/(2 sqrt(2)) cos(6 t)-(1 (1/2-(cos(6 t))/(2)))/(sqrt(2))  =  ^?sin(pi/4+3 t) (cos(3 t)-sin(3 t))

 

-(1 (1/2-(cos(6 t))/(2)))/(sqrt(2)) = (cos(6 t))/(2 sqrt(2))-1/(2 sqrt(2)):
(1)/(2 sqrt(2))+(1)/(2 sqrt(2)) cos(6 t)+(cos(6 t))/(2 sqrt(2))-(1)/(2 sqrt(2))  =  ^?sin(pi/4+3 t) (cos(3 t)-sin(3 t))

 

(1)/(2 sqrt(2))+cos(6 t) (1)/(2 sqrt(2))-(1)/(2 sqrt(2))+cos(6 t) (1)/(2 sqrt(2)) = (cos(6 t))/sqrt(2):
(cos(6 t))/sqrt(2)  =  ^?sin(pi/4+3 t) (cos(3 t)-sin(3 t))

 

sin(pi/4+3 t) = cos(3 t) sin(pi/4)+cos(pi/4) sin(3 t):
(cos(6 t))/sqrt(2)  =  ^?cos(3 t) sin(pi/4)+cos(pi/4) sin(3 t) (cos(3 t)-sin(3 t))

 

sin(pi/4) = 1/sqrt(2):
(cos(6 t))/sqrt(2)  =  ^?(cos(3 t)-sin(3 t)) (1/sqrt(2) cos(3 t)+cos(pi/4) sin(3 t))

 

cos(pi/4) = 1/sqrt(2):
(cos(6 t))/sqrt(2)  =  ^?(cos(3 t)-sin(3 t)) ((1 cos(3 t))/sqrt(2)+1/sqrt(2) sin(3 t))

 

(cos(3 t)-sin(3 t)) ((cos(3 t) 1)/sqrt(2)+(1 sin(3 t))/sqrt(2)) = cos(3 t)^2/sqrt(2)-sin(3 t)^2/sqrt(2):
(cos(6 t))/sqrt(2)  =  ^?cos(3 t)^2/sqrt(2)-sin(3 t)^2/sqrt(2)

 

cos(3 t)^2 = 1/2 (1+cos(6 t)):
(cos(6 t))/sqrt(2)  =  ^?1/sqrt(2) (1+cos(6 t))/2-sin(3 t)^2/sqrt(2)

 

(1+cos(6 t))/2 = 1/2+1/2 cos(6 t):
(cos(6 t))/sqrt(2)  =  ^?1/sqrt(2) 1/2+(cos(6 t))/2-sin(3 t)^2/sqrt(2)

 

sin(3 t)^2 = 1/2 (1-cos(6 t)):
(cos(6 t))/sqrt(2)  =  ^?(1 (1/2+(cos(6 t))/2))/(sqrt(2))-1/sqrt(2) (1-cos(6 t))/2

 

(1-cos(6 t))/2 = 1/2-1/2 cos(6 t):
(cos(6 t))/sqrt(2)  =  ^?(1 (1/2+(cos(6 t))/2))/(sqrt(2))-1/sqrt(2) 1/2-(cos(6 t))/(2)

 

(1 (1/2+(cos(6 t))/2))/(sqrt(2)) = 1/(2 sqrt(2))+(cos(6 t))/(2 sqrt(2)):
(cos(6 t))/sqrt(2)  =  ^?(1)/(2 sqrt(2))+(1)/(2 sqrt(2)) cos(6 t)-(1 (1/2-(cos(6 t))/(2)))/(sqrt(2))

 

-(1 (1/2-(cos(6 t))/(2)))/(sqrt(2)) = (cos(6 t))/(2 sqrt(2))-1/(2 sqrt(2)):
(cos(6 t))/sqrt(2)  =  ^?(1)/(2 sqrt(2))+(1)/(2 sqrt(2)) cos(6 t)+(cos(6 t))/(2 sqrt(2))-(1)/(2 sqrt(2))

 

(1)/(2 sqrt(2))+cos(6 t) (1)/(2 sqrt(2))-(1)/(2 sqrt(2))+cos(6 t) (1)/(2 sqrt(2)) = (cos(6 t))/sqrt(2):
(cos(6 t))/sqrt(2)  =  ^?(cos(6 t))/sqrt(2)

 

The left hand side and right hand side are identical:
Answer: |  (identity has been verified)

In general      \(\tan{(a + b)}=\frac{\tan a+\tan b}{1-\tan a\times \tan b}\)

 

Also  \(\tan{\frac{\pi}{4}}=1\)

 

So  \(\tan{(\frac{\pi}{4}+3t)}=\frac{1+\tan{3t}}{1-1\times\tan{3t}}\rightarrow \frac{1+\tan{3t}}{1-\tan{3t}}\)

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