# State whether the following are true or false. Justify your answer.(i) The value of tan A is always less than 1(ii) Sec for some value of angle A(iii) Cos A is the abbreviation used for the cosecant of angle A(iv) Cot A is the product of cot and A.(v) for some angle θ

(i) Consider a ΔABC, right-angled at B Tan A = But > 1

Tan A > 1

So, Tan A < 1 is not always true

Hence, the given statement is false

(ii) Sec A =   Let AC be 12k, AB will be 5k, where k is a positive integer

Applying Pythagoras theorem in ΔABC, we obtain

AC2 = AB2 + BC2

(12k)2 = (5k)2 + BC2

144k2 = 25k2 + BC2

BC2 = 119k2

BC = 10.9k

It can be observed that for given two sides AC = 12k and AB = 5k,

BC should be such that,

AC - AB < BC < AC + AB

12k - 5k < BC < 12k + 5k

7k < BC < 17 k

However, BC = 10.9k. Clearly, such a triangle is possible and hence, such value of sec A is possible

Hence, the given statement is true

(iii) Abbreviation used for cosecant of angle A is cosec A. And Cos A is the abbreviation used for cosine of angle A

Hence, the given statement is false

(iv) Cot A is not the product of cot and A. It is the cotangent of A

Hence, the given statement is false

(v) sin θ = In a right-angled triangle, hypotenuse is always greater than the remaining two sides. Therefore, such value of sin θ is not possible

Hence, the given statement is false

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