Question

If sinθ – cosθ = 0, then the value of (sin⁴θ + cos⁴θ) is ______.

Options

• 1

• `3/4`

• `1/2`

• `1/4`

Answer 1

If sinθ – cosθ = 0, then the value of (sin⁴θ + cos⁴θ) is `underlinebb(1/2)`.

Explanation:

Given,

sin θ – cos θ = 0

⇒ sin θ = cos θ

⇒ `sintheta/costheta` = 1

⇒ tan θ = 1   ...`[∵ tan theta = sintheta/costheta  "and"  tan 45^circ = 1]`

⇒ tan θ = tan 45°

∴ θ = 45°

Now, sin⁴θ + cos⁴θ = sin⁴45° + cos⁴45°

= `(1/sqrt(2))^4 + (1/sqrt(2))^4`    ...`[∵ sin 45^circ = cos 45^circ = 1/sqrt(2)]`

= `1/4 + 1/4`

= `2/4`

= `1/2`

Answer 2

LHS =`sin theta / ((1+costheta))+((1+costheta))/sin theta`

       =`(sin^2 theta +(1 +cos theta)^2)/((1+cos theta)sin theta)`

       =`(sin ^2 theta +1+cos^2theta+2costheta)/((1+cos theta)sintheta)`

       =`(1+1+2 cos theta)/((1+cos theta )sin theta)`

      =`(2+2 cos theta)/((1+cos theta )sintheta)`

       =`(2(1 + cos theta))/((1+ cos theta)sin theta)`

        =`2/sin theta`

        =`2 cosec  theta` 

       = RHS
    Hence, L.H.S = R.H.S.