Given sin `theta` +cos `theta` =1, calculate tg 2`theta` ?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to use the following trigonometric identity to evaluate ` tan 2 theta` , such that:

`tan 2theta = (sin 2 theta)/(cos 2 theta)`

The problem provides the following relation, such that:

`sin theta + cos theta = 1`

Squaring both sides, yields:

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

Expanding the square, yields:

`sin^2 theta + cos^2 theta + 2sin theta*cos theta = 1`

Using the basic trigonometric identity `sin^2 theta + cos^2 theta = 1` , yields:

`1 + 2sin theta*cos theta = 1`

Isolating `2sin theta*cos theta` to one side yields:

`2sin theta*cos theta = 1 - 1 => 2sin theta*cos theta = 0`

Using the double angle identity yields:

`2sin theta*cos theta = sin 2 theta`

Replacing `sin 2 theta` for `2sin theta*cos theta` yields:

`sin 2 theta = 0`

Replacing 0 for `sin 2 theta` in the identity that helps you to evaluate `tan 2 theta` , yields:

`tan 2theta = 0/(cos 2 theta) = 0`

Hence, evaluating `tan 2theta` , under the given conditions, yields `tan 2theta = 0` .

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