Hello! I need to verify this identity. Can you explain with all the steps? Thanks.

Both sides =  

Step-by-step explanation:

Let us revise some identity

sin(∝ + a) = sin(∝) cos(a) + cos(∝) sin(a)

cos(∝ + a) = cos(∝) cos(a) - sin(∝) sin(a)

tan ∝ =

Let us simplify each side

Left hand side

∵  ⇒ (1)

∵ ⇒ (2)

∵ and  

- Substitute them in (2)

∴

∴

- Take as a common factor

∴ ⇒ (3)

∵ ⇒ (4)

∵ and  

- Substitute them in (4)

∴

∴

- Take as a common factor

∴ ⇒ (5)

Substitute (3) and (5) in (1)

∴  

- Simplify it by divide up and down by

∴

∴ Left hand side =

Right hand side

Right hand side =    ⇒ (1)

∵ ⇒ (2)

∵ and  

- Substitute them in (2)

∴

∴

- Take as a common factor

∴ ⇒ (3)

∵ ⇒ (4)

∵ and  

- Substitute them in (4)

∴

- Take as a common factor

∴ ⇒ (5)

Substitute (3) and (5) in (1)

∴     =  

- Simplify it by divide up and down by

∴     =  

∴ Right hand side =

∵ cos∝ + sin∝ = sin∝ + cos∝

∴ Left hand side = Right hand side

∴ The identity is verified

answer: y=4000x+14000

step-by-step explanation: