Question

If sin theta + cos theta is root 2 sin (90 - theta) then what is the value of cos theta

Answer 1

ANSWER;

Given

sinθ + cosθ = √2 sin(90° - θ)

sinθ + cosθ = √2 cosθ

Squaring on both sides

(sinθ + cosθ)² = (√2)²(cosθ)²

Using

(a + b)² = a² + b² + 2ab

sin²θ + cos²θ + 2sinθcosθ = 2cos²θ

1 = 2cos²θ - 2sinθcosθ

2cos²θ - 2sinθcosθ - 1 = 0

2cos²θ - 1 - 2sinθcosθ = 0

We know that

2cos²θ - 1 = cos(2θ)

2sinθcosθ = sin(2θ)

cos(2θ) - sin(2θ) = 0

cos²θ - sin²θ - 2sinθcosθ = 0

(cosθ - sinθ)² = 0

cosθ - sinθ = 0

cosθ = sinθ

Hence cosθ = sinθ

MORE INFORMATION:

→ sin²θ + cos²θ = 1

→ sec²θ - tan²θ = 1

→ cosec²θ - cot²θ = 1

→ sin(2θ) = 2sinθcosθ

→ cos(2θ) = cos²θ - sin²θ

CONCEPTS USED:

→ Trigonometric ratios

→ Trigonometric identities

→ Trigonometric ratios of allied angles

→ Trigonometric ratios of multiples & submultiples