Question

if cos theta + sin theta is equal to root 2 cos theta then prove that cos theta minus sin theta equal to plus minus root under
theta

Answer 1

Hello there!!!!
i will be using A instead of theta
we have ,
cosA+sinA=√2cosA
squaring both the sides
=>(cosA+sinA)²=2cos²A
=>cos²A+sin²A+2sinAcosA=2cos²A
=>cos²A-2cos²A+2sinAcosA= -sin²A
=> -cos²A+2sinAcosA= -sin²A
=> cos²A-2sinAcosA=sin²A
adding sin²A on both the sides
=> cos²A+sin²A-2sinAcosA=2sin²A
=> (cosA-sinA)²=2sin²A
=> cosA-sinA=√2sinA

:D hope this helped you..

Answer 2

Step-by-step explanation:

We have,

→ cos θ + sin θ = √2cos θ .

[ Squaring both side, we get ] .

⇒ ( cos θ + sin θ )² = 2cos²θ .

⇒ cos²θ + sin²θ + 2cosθsinθ = 2cos² .

⇒ sin²θ + 2cosθsinθ = 2cos²θ - cos²θ .

⇒ sin²θ + 2cosθsinθ = cos²θ .

⇒ cos²θ - 2cosθsinθ = sin²θ .

[ Adding sin²θ both side, we get ] .

⇒ cos²θ - 2cosθsinθ + sin²θ = sin²θ + sin²θ .

⇒ ( cos θ - sin θ )² = 2sin²θ .

⇒ cos θ - sin θ = √( 2sin²θ ) .

∴ cos θ - sin θ = √2sin θ . .......

Hence, it is proved .