Question

Cos ² α+tg² α * ctg² α + sin²α=

Answer 1

Tgα * Ctgα = 1
Sin²α + Cos²α = 1

Cos²α + tg²α * Ctg²α + Sin²α = (Cos²α + Sin²α) + tg²α * Ctg²α = 1 + 1 = 2

Answer 2

Cos²α + tg²αctg² + sin²α = 1 + 1 = 2

Answer 3

To simplify the expression, we can use trigonometric identities. Let's break it down step by step:

1. Trigonometric identities used:

• tan^2(α) = sec^2(α) - 1
• cot^2(α) = cosec^2(α) - 1
• sin^2(α) + cos^2(α) = 1

2. Rewrite the expression using the identities: cos^2(α) + tan^2(α) * cot^2(α) + sin^2(α)

3. Substitute the identities: cos^2(α) + (sec^2(α) - 1) * (cosec^2(α) - 1) + sin^2(α)

4. Expand the expression: cos^2(α) + sec^2(α) * cosec^2(α) - sec^2(α) - cosec^2(α) + 1 + sin^2(α)

5. Use the identity sin^2(α) + cos^2(α) = 1: cos^2(α) + sec^2(α) * cosec^2(α) - sec^2(α) - cosec^2(α) + 2

6. Now, we can use the relationship between sec(α) and cosec(α): sec(α) = 1/cos(α) cosec(α) = 1/sin(α)

7. Substitute the above values into the expression: cos^2(α) + (1/cos^2(α)) * (1/sin^2(α)) - (1/cos^2(α)) - (1/sin^2(α)) + 2

8. To simplify further, let's get a common denominator for the last two terms: cos^2(α) + (1/cos^2(α)) * (1/sin^2(α)) - (sin^2(α) + cos^2(α)) / (cos^2(α) * sin^2(α)) + 2

9. Apply sin^2(α) + cos^2(α) = 1 again: cos^2(α) + (1/cos^2(α)) * (1/sin^2(α)) - 1 / (cos^2(α) * sin^2(α)) + 2

10. Now, we can combine the fractions: cos^2(α) + 1 / (cos^2(α) * sin^2(α)) - 1 / (cos^2(α) * sin^2(α)) + 2

11. The terms with the denominators cancel out: cos^2(α) + 2

So, the simplified expression is: cos^2(α) + 2