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

Solution:

It is given that

sin θ + cos θ = √3

Let us square on both sides

(sin θ + cos θ)² = 3

Using the algebraic identity

(a + b)² = a² + b² + 2ab

sin² θ + cos² θ + 2sin θ cos θ = 3

We know that

sin² θ + cos² θ = 1

1 + 2sin θ cos θ = 3

2sin θ cos θ = 3 - 1

2sin θ cos θ = 2

Divide both sides by 2

sin θ cos θ = 1 = sin² θ + cos² θ

Here

1 = (sin² θ + cos² θ)/ sin θ cos θ

So we get

tan θ + cot θ = 1

Therefore, it is proved.

✦ Try This: Prove that sec θ(1 - sinθ) (secθ + tanθ) =1

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 8

NCERT Exemplar Class 10 Maths Exercise 8.3 Sample Problem 4

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

Summary:

Trigonometric ratios are the ratios of the length of sides of a triangle. If sin θ + cos θ = √3, then it is proved that tan θ + cot θ = 1

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