Question

if sin a + cos a =1 prove that sin a - cos a = +-1

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Answer 1

Given---> Sin A + CosA = 1

To prove ---> SinA - CosA = ± 1

Proof --->

SinA + CosA = 1

Squaring both sides

( SinA + CosA )² = ( 1 )²

We have an identity

( a + b )² = a² + b² + 2ab

Applying it here we get

=> Sin²A + Cos²A + 2 SinA CosA = 1

we have an identity

Sin²θ + Cos²θ = 1

Applying it here

=> 1 + 2 SinA CosA = 1

=> 2 SinA CosA = 1 - 1

=> 2 SinA CosA = 0

Now

We have an identity

( a - b )² = a² + b² - 2 a b

(SinA - CosA)²= Sin²A+Cos²A-2SinACosA

( Using Sin²θ + Cos²θ = 1 and

2 SinA CosA = 0 here we get )

= 1 - 0

( SinA - CosA )² = 1

Taking square root of both sides

( SinA - CosA ) = ± 1

Answer 2

Answer:

Step-by-step explanation:

Using the formula

( a + b ) ²+ ( a− b )²= 2 ( a ² + b ² )

( cos θ + sin θ ) ² + ( cos θ − sin θ ) ²= 2 ( sin ² θ + cos ² θ )

⇒ 1 ² + ( cos θ -sin θ ) ² = 2 × 1

⇒ ( cos θ − sin θ ) ² = 2 − 1 =1

⇒ cos θ − sin θ

= ± 1