Q)Find the value of the expression : sin² 220+ sin² 68⁰+cos² 220 + cos² 680/ sin² 630 + cos 63º sin 29⁰+tan 10 tan 20 tan 30..... tan 89⁰
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Q)Find the value of the expression : sin² 220+ sin² 68⁰+cos² 220 + cos² 680/ sin² 630 + cos 63º sin 29⁰+tan 10 tan 20 tan 30..... tan 89⁰
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Step-by-step explanation:
To find the value of the given expression, we'll break it down step by step:
1. First, let's simplify the trigonometric identities:
- `sin²(220°)` is equal to `sin²(180° + 40°)` which is the same as `sin²(40°)` because `sin(180° + θ)` is equal to `sin(θ)`.
- `sin²(68°)` is already in the correct form.
- `cos²(220°)` is equal to `cos²(180° + 40°)` which is the same as `cos²(40°)` because `cos(180° + θ)` is equal to `cos(θ)`.
- `cos²(680°)` is equal to `cos²(180° + 360° + 140°)` which is the same as `cos²(140°)` because `cos(180° + n * 360° + θ)` is equal to `cos(θ)` for integer values of `n`.
- `sin(630°)` is the same as `sin(270° + 360°)` which is the same as `sin(270°)` because `sin(θ + 360°)` is equal to `sin(θ)`.
2. Now, let's calculate the trigonometric values for these angles using standard trigonometric identities and properties:
- `sin(40°)` can be calculated as `sin(40°) = sin(180° - 140°) = sin(140°)` because `sin(180° - θ)` is equal to `sin(θ)`.
- `sin(68°)` is already in the correct form.
- `cos(40°)` is the same as `sin(50°)` because `cos(θ)` is equal to `sin(90° - θ)`.
- `cos(140°)` is the same as `sin(50°)` because `cos(θ)` is equal to `sin(90° - θ)`.
3. Now, plug these values into the expression:
- `(sin²(40°) + sin²(68°) + cos²(40°) + sin²(50°)) / (sin²(50°) + sin(29°) * tan(10°) * tan(20°) * tan(30°) * ... * tan(89°))`
4. At this point, we have several trigonometric values and the product of tangents of various angles. You can calculate the numerical value of this expression, but it involves multiple calculations. If you need the specific numerical result, you can perform these calculations using a calculator or software that supports trigonometric functions.
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