Question

If
$cos \theta + \sin \theta = \sqrt{2} \sin(90 - \theta)$
then
$cos \: \theta - \sin\theta =$
Get 51 points for the answer​

Answer 1

Answer:

√2sin θ .

Step-by-step explanation:

→ 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 .

Answer 2

Answer:

cos@+sin@ = √2sin(90-@)

=> cos@ + sin@ = √2cos@

=> cos²@+2sin@cos@+sin²@ = 2cos²@

=> 1 + 2sin@cos@ = 2cos²@

=> 2sin@cos@ = 2cos²@-1

=> sin2@ = cos2@

=> (cos@-sin@)²

=> cos²@ - 2sin@cos@ + sin²@

=> 1 - cos2@

=> 2sin²@

=> cos@ - sin@ = √2sin@