If cos -1 and cos -1-2- where - - - then the number of ordered pairs - - satisfying both the equations is

The correct option is D 4cos (α β) =1 ⇒α β=2nπ, n∈ℤ Since nbsp;α, β∈ ( π, π), we have α β =0 ⇒α =β Now, nbsp;cos (α + β)=12 nbsp; ⇒cos 2 α =12 Clear ...

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Standard XII

Mathematics

Trigonometric Ratios of Common Angles

If cos α-β=1 ...

Question

If $cos (α - β) = 1$ and $cos (α + β) = \frac{1}{2},$ where $α, β \in (- π, π),$ then the number of ordered pairs $(α, β)$ satisfying both the equations is

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Solution

The correct option is D $4$
$cos (α - β) = 1$
$\Rightarrow α - β = 2 n π, n \in Z$
Since $α, β \in (- π, π),$ we have
$α - β = 0$
$\Rightarrow α = β$

Now, $cos (α + β) = \frac{1}{2}$
$\Rightarrow cos 2 α = \frac{1}{2}$

Clearly, $y = cos 2 x = \frac{1}{2}$ has four solutions in $(- π, π)$

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If cos(α−β)=1 and cos(α+β)=12, where α,β∈(−π,π), then the number of ordered pairs (α,β) satisfying both the equations is

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