State whether the following are true or false. Justify your answer

(i) Sin (A + B) = sin A + sin B

(ii) The value of sin θ increases as θ increases

(iii) The value of cos θ increases as θ increases.

(iv) Sin θ = cos θ for all values of θ.

(v) Cot A is not defined for A = 0°

(i) sin (A + B) = sin A + sin B

Let A = 30° and B = 60°

sin (A + B) = sin (30° + 60°)

= sin 90°

= 1

sin A + sin B = sin 30° + sin 60°

=

=

Clearly, sin (A + B) ≠sin A + sin B

Hence, the given statement is false

(ii) The value of sin θ increases as θ increases in the interval of 0° < θ < 90° as sin 0° = 0

sin 30^{o} =

sin 45^{o} = = 0.707

sin 60^{o} = = 0.886

sin 90° = 1

Hence, the given statement is true

(iii) cos 0° = 1

cos 30^{o} =

cos 45^{o} =

cos 60^{o} =

cos 90° = 0

It can be observed that the value of cos θ does not increase in the interval of 0° < θ < 90°

Hence, the given statement is false

(iv) sin θ = cos θ for all values of θ.

This is true when θ = 45°

As,

sin 45^{o} =

cos 45^{o} =

It is not true for all other values of θ

As sin 30^{o} = and cos 30^{o} =

Hence, the given statement is false

(v) cot A is not defined for A = 0°

cot A =

cot 0^{o} =

= =undefined

Hence, the given statement is true

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