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Question
If sinθ – cosθ = 0, then the value of (sin4θ + cos4θ) is ______.
Options
1
`3/4`
`1/2`
`1/4`
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Solution 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`
Solution 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.
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