Question

Prove that : tan 142 1/2= 2 +✓2–✓3- ✓6​

Answer 1

Answer:

Step-by-step explanation:

Go through following steps:

tan(142.5)=tan(180–37.5)

tan(142.5)=-tan(37.5)

tan(142.5)=-tan(75/2)

-tan(142.5)=tan(45/2+30/2)

-tan(142.5)=(tan(45/2)+tan(30/2))/(1-tan(45/2)*tan(30/2))

Now, we have tan(45/2)=((1-cos(45))/(1+cos(45)))^0.5

i.e. tan(45/2)=((2–2^0.5)/(2+2^0.5))^0.5

Also, we find tan(30/2)=((1-cos(30))/(1+cos(30)))^0.5

i.e. tan(30/2)=((2–3^0.5)/(2–3^0.5))^0.5

Substituting these values, we find

-tan(142.5)=(((2–2^0.5)/(2+2^0.5))^0.5+((2–3^0.5)/(2–3^0.5))^0.5)/(1-((2–2^0.5)/(2+2^0.5))^0.5*((2–3^0.5)/(2–3^0.5))^0.5)  

i.e. tan(142.5)=(((2–2^0.5)/(2+2^0.5))^0.5+((2–3^0.5)/(2–3^0.5))^0.5)/(-1+((2–2^0.5)/(2+2^0.5))^0.5*((2–3^0.5)/(2–3^0.5))^0.5)

i.e. tan(142.5)=(2^0.5+1–3^0.5)/(3+6^0.5–3^0.5–2^1.5)

If you rationalize the denominator, you will get

tan(142.5)=2+2^0.5–3^0.5–6^0.5