Question

If X and Y are complementary angles then show that:

a) sin^2X+sin^2Y=1​

Answer 1

Step-by-step explanation:

Given :-

X and Y are complementary angles

To find:-

Show that :sin^2X+sin^2Y=1

Solution:-

Given that :

X and Y are the complementary angles

We know that

The sum of two angles is equal to 90° then they are complementary angles.

X+Y = 90°

=> X = 90°-Y

On taking Sin both sides then

=> Sin X = Sin (90°-Y)

We know that

Sin (90°-A) = CosA

=> Sin X = Cos Y

On squaring both sides then

=> (Sin X)^2 = (Cos Y)^2

=> Sin^2 X = Cos^2 Y

We know that

Sin^2 A + Cos^2 A = 1

=> Sin^2 X = 1 - Sin^2 Y

=> Sin^2 X + Sin^2 Y = 1

Hence, Proved.

(Or)

Given that :

X and Y are the complementary angles

We know that

The sum of two angles is equal to 90° then they are complementary angles.

X+Y = 90°

=> X = 90°-Y

On taking Cos both sides then

=> Cos X = Cos (90°-Y)

We know that

Cos (90°-A) = Sin A

=> Cos X = Sin Y

On squaring both sides then

=> (Cos X)^2 = (Sin Y)^2

=> Cos^2 X = Sin^2 Y

We know that

Sin^2 A + Cos^2 A = 1

=> 1- Sin^2 X = Sin^2 Y

=> Sin^2 X + Sin^2 Y = 1

Hence, Proved.

Answer:-

If X and Y are complementary angles then Sin^2X+Sin^2Y=1

Used formulae:-

Complementary angles:-

• The sum of two angles is equal to 90° then they are complementary angles.

• Cos (90°-A) = Sin A

• Sin (90°-A) = CosA

• Sin^2 A + Cos^2 A = 1