If cos-cos-0-sin-sin- then which of the following statements is-are true-

The correct option is D sin^2 ( α+π/2 )+sin^2 ( β+π/2 )=1 cos(α+β) cosα+cos β = 0 …… (1) sin α + sin β = 0 …… (2) (1)^2+(2)^2⇒ 2+2 cos(α β)=0 ⇒ cos(α β)= 1 ( ...

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Byju's Answer

Standard XII

Mathematics

Trigonometric Ratios of Allied Angles

If cosα+cosβ=...

Question

If $c o s α + c o s β = 0 = s i n α + s i n β$ , then which of the following statement(s) is/are true?

c o s (α - β) = - 1

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c o s 2 α + c o s 2 β + 2 c o s (α + β) = 0

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s i n 2 α + s i n 2 β + 2 s i n (α + β) = 0

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s i n^{2} (α + \frac{π}{2}) + s i n^{2} (β + \frac{π}{2}) = 1 - c o s (α + β)

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Solution

The correct options are
A $c o s (α - β) = - 1$
B $c o s 2 α + c o s 2 β + 2 c o s (α + β) = 0$
C $s i n 2 α + s i n 2 β + 2 s i n (α + β) = 0$
D $s i n^{2} (α + \frac{π}{2}) + s i n^{2} (β + \frac{π}{2}) = 1 - c o s (α + β)$

$c o s α + c o s β = 0 \dots \dots (1)$
$s i n α + s i n β = 0 \dots \dots (2)$
$(1)^{2} + (2)^{2} \Rightarrow 2 + 2 c o s (α - β) = 0 \Rightarrow c o s (α - β) = - 1$
$(1)^{2} - (2)^{2} \Rightarrow c o s 2 α + c o s 2 β + 2 c o s (α + β) = 0$
$((1) - (2))^{2} \Rightarrow (c o s α + c o s β + s i n α + s i n β)^{2} = 0$
$\Rightarrow 2 + 2 c o s (α - β) + 2 c o s α s i n α + 2 c o s β s i n β + 2 c o s β s i n α = 0$
Using $c o s (α - β) = - 1$ we get
$2 - 2 + s i n 2 α + s i n 2 β + 2 s i n (α + β) = 0$
$s i n 2 α + s i n 2 β + 2 s i n (α + β) = 0$
Lastly,
$s i n^{2} (α + \frac{π}{2}) + s i n^{2} (β + \frac{π}{2}) = c o s^{2} α + c o s^{2} β$
$= 2 c o s^{2} α = 2 c o s^{2} β [u s i n g | c o s α | = | c o s β |]$
$= 1 + c o s 2 α = 1 + c o s 2 β$
$= 1 + \frac{c o s 2 α + c o s 2 β}{2}$
$= 1 - c o s (α + β)$ [using result in (B)]

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c o s α + c o s β + c o s γ = s i n α + s i n β + s i n γ = 0

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c o s (β + γ) + c o s (γ + α) + c o s (α + β) = 0

s i n (β + γ) s i n (γ + α) + s i n (α + β) = 0

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sin α + sin β = 0 = cos α + cos β

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Q. If

sin α + sin β + sin γ = 0

and

cos α + cos β + cos γ = 0

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cos (α - β) + cos (β - γ) + cos (γ + α)

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c o s α + c o s β + c o s γ = s i n α + s i n β + s i n γ = 0

, then :

c o s 2 α + c o s 2 β + c o s 2 γ = s i n 2 α + s i n 2 β + s i n 2 γ = 0

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c o s (β - γ) + c o s (γ - α) + c o s (α - β) = - \frac{3}{2}

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c o s α + c o s β + c o s γ = 0

and

s i n α + s i n β + s i n γ = 0

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If cos α+cos β=0=sin α+sinβ, then which of the following statement(s) is/are true?

If $c o s α + c o s β = 0 = s i n α + s i n β$ , then which of the following statement(s) is/are true?