Question

if cos theta + sin theta is equal to root 2 cos theta then prove that cos theta minus sin theta is equal to root 2 Sin Theta

Answer 1

Heya User ,

--> cos θ + sin θ = √2 cos θ
--> sin θ = √2 cos θ - cos θ
=> sin θ = ( √2 - 1 ) cos θ

=> [ sin θ / ( √2 - 1 ) ] = cos θ
=> [ sin θ ( √2 + 1 ) / ( 2 - 1 ) ] = cos θ

0_0 --> We rationalized the denominator in the 2nd step ^_^ 

=> [ √2 sin θ + sin θ ] = cos θ
=> cos θ - sin θ = √2 sin θ

Answer 2

Question :-

→ If cos θ + sin θ = √2cos θ , then prove that cos θ - sin θ = √2sin θ .

Answer :-

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 .