Question

if cos-sin=1, then show that cos+sin=1 or -1

Answer 1

♧♧HERE IS YOUR ANSWER♧♧

Trigonometry is the study of angles and its ratios. This study prescribes the relation amongst the sides of a triangle and angles of the triangle.

There are sine, cosine, tangent, cot, sectant and cosec ratios to an angle of a triangle.

Let me tell you an interesting fact about Trigonometry.

"Triangle" > "Trigonometry"

Remember some formulae now :

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1

Want to learn more!

Here it is :

sin(A + B) = sinA cosB + cosA sinB

sin(A - B) = sinA cosB - cosA sinB

cos(A + B) = cosA cosB - sinA sinB

cos(A - B) = cosA cosB + sinA sinB

SOLUTION :

Given,

cosθ - sinθ = 1

Squaring both sides, we get

(cosθ - sinθ)² = 1²

=> cos²θ + sin²θ - 2 sinθ cosθ = 1

=> 1 - 2 sinθ cosθ = 1

=> 2 sinθ cosθ = 0 .....(i)

Now,

(cosθ + sinθ)²

= cos²θ + sin²θ + 2 sinθ cosθ

= 1 + 0, by (i)

= 1

So, cosθ + sinθ = ± 1

Hence,

cosθ + sinθ = 1 or -1

Therefore, proved.

♧♧HOPE IT HELPS YOU♧♧