Question

Question #696f9

Answer 1

Prove trig identity

Explanation:

Use these trig identities:
`1. tan (a + b) = (tan a + tan b)/(1 - tan a.tan b)`
`2. 1 + cos 2a = 2cos^2 a`
3. sin 2a = 2sin a.cos a
Apply the first trig identity to the left side of the equation:
`LS = tan (45 + tan (t/2)) = (1 + tan (t/2))/(1 - tan (t/2)) =`
Replace `tan (t/2)` by `(sin t/2)/(cos t/2)`, we get:
`LS = (cos (t/2) + sin (t/2))/(cos (t/2) - sin (t/2)=` (1)
Now, transform the right side of the equation:
`RS = (1 + cos t + sin t)/( 1 + cos t - sin t) =`
Replace
`1 + cos t` by `2cos^2 (t/2)`
`sin t = 2sin (t/2).cos (t/2)`, we get
`RS = (2cos^2 (t/2)(cos (t/2) + sin (t/2)))/(2cos^2 (t/2)(cos (t/2) - sin (t/2)`.
`RS = (cos (t/2) + sin (t/2))/(cos (t/2) - sin (t/2)) = LS`
The trig identity is proven.