Question

if x sin cube theta + Y cos cube theta is equal to sin theta into cos theta and x sin theta is equal to Y cos theta show X square + Y square is equal to 1​

Answer 1

Given

→ x sin³∅ + y cos³∅ = sin∅ cos∅

→ x sin∅ = y cos∅

Solution

→ x sin³∅ + y cos³∅ = sin∅ cos∅

Taking common term out.

→ x sin∅(sin²∅) + y cos∅(cos²∅) = sin∅ cos∅

By replacing x sin∅ by y cos∅,

→ y cos∅(sin²∅) + y cos∅(cos²∅) = sin∅ cos∅

→ y cos∅(sin²∅ + cos²∅) = sin∅ cos∅

Since sin²∅ + cos²∅ = 1

→ y cos∅ = sin∅ cos∅

→ y = sin∅

Now by using this value

→ x sin∅ = y cos∅

→ x × y = y cos∅

→ x = cos∅

Since cos²∅ + sin²∅ = 1 therefore

→ x² + y² = 1

Hence Proved

Answer 2

Answer:

Given,

x sin³ ¢ + y cos³¢ = sin ¢. cos ¢

x sin ¢ = y cos ¢ --> ( i )

Proving :

sin²¢. x sin ¢ + cos²¢. y cos ¢ = sin ¢. cos ¢

sin²¢. y cos ¢ + cos²¢. y cos ¢ = sin ¢. cos ¢

y cos ¢ ( sin²¢ + cos²¢) = sin ¢. cos ¢

y cos ¢ = sin ¢. cos¢

y = sin ¢ --> ( a )

Now, Putting value of 'y' in (i),

x sin ¢ = sin ¢. cos ¢

x = cos ¢ --> ( b )

Adding and Squaring ( a ) and ( b ),

x² + y² = cos² ¢ + sin² ¢

Hence,

x² + y² = 1

PROVED.