Question

If cos(alpha + beta) + sin(alpha – beta) = 0. then (alpha + beta) can be -

1. π
2. 3π/4
3. 5π
4. 5π/2​

Answer 1

Required Answer:-

Cos (α + β) + sin(α - β) = 0

Now, I will suggest to expand these trigonometric ratios separately by using identities. The identities are:

Identities that can be used:-

cos (a + b) = cos a cos b - sin a sin b --- (1)

cos (a - b) = cos a cos b + sin a sin b --- (2)

sin (a + b) = sin a cos b + cos a sin b ---(3)

sin (a - b) = sin a cos b - cos a sin b ---(4)

Now putting the required identities (1) and (4) with respectives α and β as given in the question.

➙ Cos (α + β) + sin(α - β) = 0

➙ Cos α cos β - Sin α sin β + sin α cos β - cos α sin β = 0

➙ Cos α cos β + sin α cos β - sin α sin β - cos α sin β = 0

Taking cos β from the first two terms and sin β in the next two terms as common.

➙ cos β (Cos α + sin α) - sin β (Sin α + cos α) = 0

➙ (cos β - sin β)(cos α + sin α) = 0

Then,

If cos β - sin β = 0

⇒ cos β = sin β

⇒ tan β = 1

Assuming β to be in range [0, 180]

⇒ β = 45°

And,

if cos α + sin α = 0

⇒ cos α = - sin α

⇒ tan α = -1

Assuming α to be in range [0, 180]

⇒ α = 135°

Then, α + β = 135° + 45° = 180°/π radians

Option A is the correct option which also indicates that our assumption wasn't wrong earlier.