If sinθ+cosθ=√3, then prove that . . .

Question : If sinθ+cosθ=√3, then prove that tanθ+cotθ=1

CBSE Sample Question Paper 2025

Solution : 

sinθ+cosθ=√3
S.B.S
(sinθ+cosθ)²=(√3)²
sin²θ+cos²θ+2sinθcosθ=3
1+2sinθcosθ=3 [∵sin²θ+cos²θ=1]
2sinθcosθ=3-1
2sinθcosθ=2
sinθcosθ=2/2
sinθcosθ=1 — (1) 

To Prove : 
tanθ+cotθ=1

Proof : 
LHS:

tanθ+cotθ
=sinθ/cosθ + cosθ/sinθ
=[sin²θ+cos²θ]/[sinθcosθ]
=1/[sinθcosθ] [∵sin²θ+cos²θ=1]
= 1/1 [From Equation (1)]
=1
=RHS

LHS=RHS
Hence Proved.