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Question
State whether the following are true or false. Justify your answer.
The value of tan A is always less than 1.
Options
True
False
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Solution
This statement is False.
Explanation:
Consider a ΔABC, right-angled at B.
tan A = `("Side opposite to ∠A")/("Side adjacent to ∠A")`
= `12/5`
But `12/5 > 1`
∴ tan A > 1
So, tan A < 1 is not always true.
Hence, the given statement is false.
A tangent of an angle is the ratio of sides other than hypotenuse, which may be equal or unequal to each other.
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Prove that: cot θ + tan θ = cosec θ·sec θ
Proof: L.H.S. = cot θ + tan θ
= `square/square + square/square` ......`[∵ cot θ = square/square, tan θ = square/square]`
= `(square + square)/(square xx square)` .....`[∵ square + square = 1]`
= `1/(square xx square)`
= `1/square xx 1/square`
= cosec θ·sec θ ......`[∵ "cosec" θ = 1/square, sec θ = 1/square]`
= R.H.S.
∴ L.H.S. = R.H.S.
∴ cot θ + tan θ = cosec·sec θ
