Question

The above answer is correct ?
If wrong mention the correct answer .​

Answer 1

Answer:

I think there is some think which i can't understand in this Q

Step-by-step explanation:

In 2nd last step what u did with 2absin©cos©

if y =acos© - bsin© is wriiten in place of acos©+bsin© then we can solve questions but i think there is some mistake

Maybe I'm wrong please check questions once again

Answer 2

$\large\underline{\sf{Given \: appropriate\:Question - }}$

If

$\sf \: x = asin\theta + bcos\theta \: and \: y = acos\theta - bsin\theta ,$

$\sf \: prove \: that \: \: \: {x}^{2} + {y}^{2} = {a}^{2} + {b}^{2}$

$\red{\large\underline{\sf{Solution-}}}$

Given that,

$\rm :\longmapsto\:x = asin\theta + bcos\theta$

So,

$\rm :\longmapsto\: {x}^{2} = ({ asin\theta + bcos\theta})^{2}$

$\rm :\longmapsto\: {x}^{2} = {(asin\theta )}^{2} + {(bsin\theta )}^{2} + 2(asin\theta )(bcos\theta )$

$\rm :\longmapsto\: {x}^{2} = {a}^{2} {sin}^{2}\theta + {b}^{2} {cos}^{2}\theta + 2absin\theta cos\theta - - - (1)$

Now, Given that

$\rm :\longmapsto\:y = acos\theta - bsin\theta$

So,

$\rm :\longmapsto\: {y}^{2} = {(acos\theta - bsin\theta )}^{2}$

$\rm :\longmapsto\: {y}^{2} = {(acos\theta )}^{2} + {(bsin\theta )}^{2} - 2(acos\theta )(bsin\theta )$

$\rm :\longmapsto\: {y}^{2} = {a}^{2} {cos}^{2}\theta + {b}^{2} {sin}^{2}\theta - 2absin\theta cos\theta - - - (2)$

So,

$\red{\rm :\longmapsto\: {x}^{2} + {y}^{2}}$

$\rm \:= {a}^{2} {sin}^{2}\theta + {b}^{2} {cos}^{2}\theta + 2absin\theta cos\theta + {a}^{2} {cos}^{2}\theta + {b}^{2} {sin}^{2}\theta - 2absin\theta cos\theta$

$\rm \:= {a}^{2} {sin}^{2}\theta + {b}^{2} {cos}^{2}\theta + {a}^{2} {cos}^{2}\theta + {b}^{2} {sin}^{2}\theta$

$\rm \:= {a}^{2} {sin}^{2}\theta + {a}^{2} {cos}^{2}\theta + {b}^{2} {cos}^{2}\theta + {b}^{2} {sin}^{2}\theta$

$\rm \:= {a}^{2} ({sin}^{2}\theta + {cos}^{2}\theta)+{b}^{2}({cos}^{2}\theta+ {sin}^{2}\theta)$

$\rm \: = \: {a}^{2} + {b}^{2}$

Hence,

$\red{ \boxed{ \bf \: {x}^{2} + {y}^{2} = \: {a}^{2} + {b}^{2} }}$

Additional Information:-

Relationship between sides and T ratios

sin θ = Opposite Side/Hypotenuse

cos θ = Adjacent Side/Hypotenuse

tan θ = Opposite Side/Adjacent Side

sec θ = Hypotenuse/Adjacent Side

cosec θ = Hypotenuse/Opposite Side

cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Co-function Identities

sin (90°−x) = cos x

cos (90°−x) = sin x

tan (90°−x) = cot x

cot (90°−x) = tan x

sec (90°−x) = cosec x

cosec (90°−x) = sec x

Fundamental Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1