Question

sin theta + cos theta is equal to root 3 then prove that tan theta + cot theta is equal to 1

Answer 1

for convenience , I am considering  theta equal to x 
sin x + cos x = root3 
Squaring both sides we get, 
sin^2 x + cos^2 x + 2sinxcosx = 3
Since sin^2 x + cos^x =1 
Therefore
1 + 2sinxcosx =3 
2 sinxcosx =2
sinxcosx = 1    .....(1)
Now 
tanx + cotx can be written as 
sin x/cosx + cos x/sinx
= (sin^2x + cos^2x)/sinxcosx
= 1/sinxcosx 
= 1/1 (From 1)
=1 
hence proved 
 
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Answer 2

Answer:

sin∅ + cos∅ = √3

Squaring both sides,

sin²∅ + cos²∅ + 2 sin∅cos∅ = 3

1 +  2 sin∅cos∅ = 3

2 sin∅cos∅ = 3-1

2 sin∅cos∅ = 2

sin∅cos∅ = 2/2

sin∅cos∅ = 1 .........................(1)

= tan∅ + cot∅

= sin∅/cos∅  + cos∅/sin∅

= sin²∅ +cos²∅/sin∅cos∅

=1/1    ........................(from 1)

= 1

∴ tan∅ + cot∅ = 1

hence, proved.