Question

Cos (π + 2α) + sin(π + 2α) tg(π/2 + α)

Answer 1

Cos(П + 2a) + Sin(П + 2a) * tg(П/2 + a) = - Cos2a - Sin2a * ( - ctga) =
= - Cos2a + 2Sina* Cosa * (Cosa/Sina) = - Cos2a + 2Cos²a =
= - ( Cos²a - Sin²a) + 2Cos²a = - Cos²a + Sin²a + 2Cos²a = Sin²a +Cos²a = 1

Answer 2

Let's break down the expression step by step:

1. Cosine and Sine of π + 2α: Using the angle sum formula for cosine and sine, we have:

   • cos(π + 2α) = cos(π)cos(2α) - sin(π)sin(2α) = -cos(2α)
   • sin(π + 2α) = sin(π)cos(2α) + cos(π)sin(2α) = -sin(2α)

2. Tangent of π/2 + α: The tangent of an angle is the sine of the angle divided by the cosine of the angle. In this case:

   • tg(π/2 + α) = sin(π/2 + α) / cos(π/2 + α)
   • Applying angle sum formulas:
   • tg(π/2 + α) = cos(α) / sin(α) = 1 / tan(α)

Putting it all together:

cos(π + 2α) + sin(π + 2α) tg(π/2 + α) = -cos(2α) + (-sin(2α)) * (1 / tan(α))

Simplifying further:

• cos(2α) - sin(2α) / tan(α)

This expression combines trigonometric functions of angle α. Depending on the specific value of α, you can substitute the numerical value of α to obtain a numerical result.