If 𝑐𝑜𝑠 (α + β) = 0, then 𝑠𝑖𝑛 (α − β) can be reduced to - Sri Chaitanya Infinity Learn Best Online Courses for NCERT Solutions, CBSE, ICSE, JEE, NEET, Olympiad and Class 6 to 12

We have been given that 𝑐𝑜𝑠 (α + β) = 0.

We know that 𝑐𝑜𝑠 90° = 0

⇒ 𝑐𝑜𝑠 (α + β) = 𝑐𝑜𝑠 90°

⇒ α + β = 90°

⇒ α = 90° − β

Substituting the value of α from the above equation, we get

Now, 𝑠𝑖𝑛 (α − β) = 𝑠𝑖𝑛 (90° − β − β)

⇒ 𝑠𝑖𝑛 (α − β) = 𝑠𝑖𝑛 (90° − 2β)

We know that, for any angle θ, 𝑠𝑖𝑛 (90° − θ) = 𝑐𝑜𝑠 θ

Therefore we can say that, 𝑠𝑖𝑛 (90° − 2β) = 𝑐𝑜𝑠 2β

Hence, here 𝑠𝑖𝑛 (α − β) can be reduced to 𝑐𝑜𝑠 2β.

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