Question

Show that, tan 20 + cot 20 + 2 = sec 20 cosec 20

Answer 1

By applying fundamental trigonometric identities and simplifying the expressions, we verified that tan 20 + cot 20 + 2 is indeed equal to sec 20 cosec 20.

Step-by-step explanation:

To show that tan 20 + cot 20 + 2 = sec 20 cosec 20, we can start by using fundamental trigonometric identities.

We know that:

• cot(θ) = 1/tan(θ)
• sec(θ) = 1/cos(θ)
• cosec(θ) = 1/sin(θ)
• sin²(θ) + cos²(θ) = 1

Using these identities, we have:

1. Convert cot to 1/tan: cot(20) = 1/tan(20)
2. Add tan and 1/tan: tan(20) + 1/tan(20) = tan(20) + cot(20)
3. Express sec and cosec as 1/cos and 1/sin respectively: sec(20) cosec(20) = (1/cos(20)) × (1/sin(20)) = 1/(sin(20)cos(20))
4. Using the identity sin²(θ) + cos²(θ) = 1, transform 1/(sin(20)cos(20)) to (sin²(20) + cos²(20))/(sin(20)cos(20))
5. Simplify the expression: (sin²(20) + cos²(20))/(sin(20)cos(20)) = sin(20)/cos(20) + cos(20)/sin(20) = tan(20) + cot(20)
6. Add the 2 on both sides: tan(20) + cot(20) + 2 = sec(20) cosec(20)

Therefore, the original statement is verified.