Question

If sin theta + cosec theta = 1/2, then the value of sin4 + cos4​

Answer 1

sin⁴θ + cos⁴θ = 1

it is given that, sinθ + cosecθ = 1/2

then we have to find the value of sin⁴θ + cos⁴θ

first of all, sinθ + cosecθ ≥ 1 or, ≤ -1.

given statement is wrong.

it should be,

sinθ + cosecθ = 2 ( in place of 1/2)

so, sinθ + 1/sinθ = 2

⇒sin²θ - 2sinθ + 1 = 0

⇒(sinθ - 1)² = 0

⇒sinθ = 1

and we know, if sinθ =1 then, cosθ = 0

because, sin²x + cos²x = 1 [ trigonometric identity ]

then, sin⁴θ + cos⁴θ = (1)⁴ + (0)⁴ = 1

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Answer 2

Correct Question :-------- sin@ + cosec@ = 2

we have to Find = sin⁴@ + cos⁴@ ?

we know that,

cosec@ = 1/sin@

so,

sin@ + 1/sin@ = 2

sin²@ +1 = 2sin@

sin²@ -2sin@ +1 = 0

a² - 2ab + b² = (a-b)²

so, we get,

(sin@-1)² = 0

sin@ = 1

That means in

sin²@ + cos²@ = 1

we have cos²@ = 0

so,

sin⁴@ + cos⁴@ = (1)⁴ + (0)⁴ = 1 (Ans)

Remember :-------

when sinx + cosecx = 2

sin^n x + cosec^n x = 2 (Always)