if tan 0 = 1 then cosec² 9 0 √5 2 sec² 0 Cose c² + sec² B
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if tan 0 = 1 then cosec² 9 0 √5 2 sec² 0 Cose c² + sec² B
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To solve the given task, let's break it down step by step and make sure to explain the concepts along the way, so it is understandable to a schoolchild.
Given:
tan(θ) = 1
We are asked to find the value of:
cosec²(90°) √5 + 2sec²(0°) cosec²(θ) + sec²(θ)
Step 1: Evaluating cosec²(90°)
cosec(90°) represents the cosecant function at an angle of 90 degrees, which is the reciprocal of the sine function at that angle.
The sine of 90 degrees is equal to 1, so the cosecant function will be the reciprocal of 1, which is also 1.
Therefore, cosec(90°) = 1
And cosec²(90°) = 1² = 1
Step 2: Evaluating sec²(0°)
sec(0°) represents the secant function at an angle of 0 degrees. The secant function is the reciprocal of the cosine function.
The cosine of 0 degrees is equal to 1, so the secant function will be the reciprocal of 1, which is also 1.
Therefore, sec(0°) = 1
And sec²(0°) = 1² = 1
Step 3: Evaluating cosec²(θ) + sec²(θ)
Since we are given tan(θ) = 1, we can use the following trigonometric identity:
tan²(θ) + 1 = sec²(θ)
Substituting the given information, we have:
1² + 1 = sec²(θ)
2 = sec²(θ)
Step 4: Substituting the values into the original expression
Now that we have calculated the values for cosec²(90°), sec²(0°), and sec²(θ), we can substitute them into the original expression:
cosec²(90°) √5 + 2sec²(0°) cosec²(θ) + sec²(θ)
= 1 √5 + 2(1) (2) + 2
= √5 + 4 + 2
= √5 + 6
So, the value of the given expression is √5 + 6.
Note: the value of θ was not specifically provided in the task. However, we were able to solve the expression using the given information and the trigonometric identities.