If sin θ + cos θ = √3, then prove that tan θ + cot θ = 1

Q) If sin θ + cos θ = √3, then prove that tan θ + cot θ = 1

Q29 A – Sample Question Paper – Set 1 – Maths Standard – CBSE 2026

Ans:

Given: sin θ + cos θ = √3

Step 1: By squaring on both sides, we get:

∴ (sin θ + cos θ)² = (√3)²

∴ sin² θ + cos² θ + 2 sin θ cos θ = 3

∵ sin² θ + cos² θ = 1

∴ 1 + 2 sin θ cos θ = 3

∴ 2 sin θ cos θ = 2

∴ sin θ cos θ = 1 ……….. (i)

Step 2: We need to prove that tan θ + cot θ = 1

Let’s start from LHS:

LHS = tan θ + cot θ

= $\frac{sin \theta}{cos \theta} + \frac{cos \theta}{sin \theta}$

= $\frac{(sin \theta)(sin \theta) + (cos \theta)(cos \theta)}{(cos \theta)(sin \theta)}$

= $\frac{(sin ^2 \theta + cos^2 \theta)}{sin \theta cos \theta}$

∵ sin² θ + cos² θ = 1

∴ LHS = $\frac{1}{sin \theta cos \theta}$

Step 3: By substituting value of sin θ . cos θ from equation (i), we get:

LHS = 1 / 1

= 1 = RHS

Hence Proved!

Please press the “Heart”, if you like the solution.

Contact us