Question

Chapter: Introduction to trigonometry​

Answer 1

Note : Theta is written as A.

Answer:

x^2 + y^2 = a^2 + b^2

Step-by-step explanation:

Given value of x in terms of trigonometric terms is a cosA - b sinA and y is a sinA + b cosA

Square on both sides of x :

= > x^2 = ( a cosA - b sinA )^2

= > x^2 = ( a cosA ) + ( b sinA ) - 2( a cosA x b sinA ) { using ( a - b )^2 = a^2 + b^2 - 2ab }

= > x^2 = a^2 cos^2 A + b^2 sin^2 A - 2ab cosA sinA

Square on both sides of y :

= > y^2 = ( a sinA + b cosA )^2

= > y^2 = ( a sinA )^2 + ( b cosA )^2 + 2( a sinA x b cosA ) { ( a + b )^2 = a^2 + b^2 + 2ab }

= > y^2 = a^2 sin^2 A + b^2 cos^2 A + 2ab sinA cosA

Then, adding the square of x and square of y :

= > x^2 + y^2 = [ a^2 cos^2 A + b^2 sin^2 A - 2ab cosA sinA ] + [ a^2 sin^2 A + b^2 cos^2 A + 2ab sinA cosA ]

= > x^2 + y^2 = a^2 cos^2 A + b^2 sin^2 A - 2ab sinAcosA + a^2 sin^2 A + b^2 cos^2 A + 2ab sinAcosA

= > x^2 + y^2 a^2( sin^2 A + cos^2 A ) + b^2( sin^2 A + cos^2 A )

From the properties of trigonometric ratios : sin²∅ + cos²∅ = 1

= > x^2 + y^2 = a^2( 1 ) + b^2( 1 )

= > x^2 + y^2 = a^2 + b^2

Hence proved.

Answer 2

Hi there !!

Identity used :-

(a-b) ²= a²+b²-2ab

(a+b) ² =a²+b²+2ab

sin²0 + cos²0 = 1

{0= Thetha }

Look at the attachment ↑↑

Thank you :)