Question

If `sin(theta)+cos(theta)=(sqrt2) sin(90^@-theta)`,find `tan(theta)`?

Answer 1

`tan(theta)=sqrt2 -1`

Explanation:

Given: `sin(theta)+cos(theta)=(sqrt2) sin(90^@-theta)`

Use the identitity `sin(A-B) = sin(A)cos(B)-cos(A)sin(B)` where `A = 90^@` and `B = theta`

`sin(theta)+cos(theta)=(sqrt2)(sin(90^@)cos(theta)-cos(90^@)sin(theta))`

Use the facts that `sin(90^@) = 1 and cos(90^@)=0`

`sin(theta)+cos(theta)=(sqrt2)cos(theta)`

Divide both sides of the equation by `cos(theta)`:

`sin(theta)/cos(theta)+ 1=sqrt2`

Use the identity `sin(theta)/cos(theta) = tan(theta)`

`tan(theta)+ 1=sqrt2`

Subtract 1 from both sides:

`tan(theta)=sqrt2 -1`