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

(Q 29 A – 30/3/3 – CBSE 2026 Question Paper)

Ans:

Step 1: Its given that sin θ + cos θ = √3

Squaring on both sides, we get

∴ (sin θ + cos θ) ² = (√3) ² = 3

By applying algebraic identity: (a + b) ² = a ² + b ² + 2 a b, we get:

∴ sin ² θ + cos ² θ + 2 sin θ . cos θ = 3

By trigonometric identity we know that sin ² θ + cos ² θ = 1

∴ sin ² θ + cos ² θ + 2 sin θ . cos θ = 3

∴ 1 + 2 sin θ . cos θ = 3

∴ 2 sin θ . cos θ = 3 – 1 = 2

∴ sin θ . cos θ = 1

Step 2: Next, we need to prove: tan θ + cot θ = 1

Let’s start from LHS:

∴ LHS =

∴ LHS =

Step 3: Again by trigonometric identity, we have sin ² θ + cos ² θ = 1

and we calculated value of sin θ . cos θ = 1 from step 1

By substituting these values in LHS, we get:

∴ LHS =

∴ LHS = = 1 = RHS

Hence Proved !

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