Question

if sinα + sinβ = 2
then find the value of
cos²α + cos²β​

Answer 1

EXPLANATION.

⇒ sinα + sinβ = 2.

As we know that,

If both are equal then,

⇒ sinα = 2.

⇒ sinβ = 2.

Only if,

⇒ sinα = 90°.

⇒ sinβ = 90°.

⇒ sin(90°) + sin(90°).

⇒ 1 + 1 = 2.

To find value of,

⇒ cos²α + cos²β.

Put the value of α = β = 90° in equation, we get.

⇒ cos²(90°) + cos²(90°).

⇒ 0.

                                                                                                                   

MORE INFORMATION.

Fundamental trigonometric identities.

(1) = sin²∅ + cos²∅ = 1.

(2) = 1 + tan²∅ = sec²∅.

(3) = 1 + cot²∅ = cosec²∅.

The greatest & least value of the expression [ a sin∅ + b cos∅].

Greatest value = √a² + b².

Least value = -√a² + b².

Answer 2

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${\large{\bold{\rm{\underline{Given \; that}}}}}$

${\sf{\bigstar \: sin \alpha \: + sin \beta \: = 2}}$

${\large{\bold{\rm{\underline{To \; find}}}}}$

${\sf{\bigstar \: cos^{2} \alpha \: + cos^{2} \beta}}$

${\large{\bold{\rm{\underline{Solution}}}}}$

${\sf{\bigstar \: cos^{2} \alpha \: + cos^{2} \beta \: = 0}}$

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${\large{\bold{\rm{\underline{Full \; Solution}}}}}$

~ As it's given that ${\red{sin \alpha \: + sin \beta \: = 2}}$. Henceforth,

${\sf{:\implies \: sin \alpha \: = 2}}$

${\sf{:\implies \: sin \beta \: = 2}}$

~ That's why,

${\sf{:\implies \: sin \alpha \: = 90 \degree}}$

${\sf{:\implies \: sin \beta \: = 90 \degree}}$

~ Putting the values,

${\sf{:\implies \: sin(90 \degree) + \: sin(90 \degree)}}$

${\sf{:\implies \: 1 + 1}}$

${\sf{:\implies \: 2}}$

~ As we know that we have to find ${\red{cos^{2} \alpha \: + cos^{2} \beta}}$

~ Let's put the values,

${\sf{\implies \: cos^{2}(90 \degree) \: + cos^{2} (90 \degree)}}$

${\sf{\implies \: 0}}$

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