Verify the following equalities.

(i) sin²30° + cos²30° = 1

(ii) 1 + tan²45° = sec²45°

(iii) cos 60° = 1 – 2sin² 30° = 2cos² 30° – 1

(iv) cos 90° = 1 – 2 sin² 45° = 2cos² 45° – 1

(v)

(vi)

(vii)

(viii) tan²60° – 2tan²45° – cot²30° + 2sin²

(ix) 4cot²45° – sec²60° + sin² 60° = 1

(x) sin30° cos60° + cos30° sin60° = sin 90°

(i) We know sin 30° = (1/2) and cos 30° = (√3/2) (From the table)

Putting the values in the left hand side:

⇒

⇒

⇒ 1

Which is equal to the right hand side.

Hence verified.

(ii) We know tan 45° = 1 and sec 45° = √2 (From the table)

Putting the values in the left hand side:

⇒ 1 + (1)²

⇒ 1 + 1

⇒ 2

Putting the values in the right hand side:

⇒ (√2)²

⇒ 2

∴ LHS = RHS

Hence verified.

(iii) We know sin 30° = cos 60° = (1/2) and cos 30° = (√3/2) (From the table)

So the leftmost function = cos60° = (1/2)

Putting values in the middle function:
⇒ 1 – 2×

⇒

⇒ 1/2

Putting values in the rightmost function:
⇒ 2×

⇒

⇒ 1/2

Therefore, all simplify to 1/2 and are equal.

Hence verified.

(iv) We know sin 45° = cos 45° = (1/√ 2) and cos 90° = 0 (From the table)

So the leftmost function = cos90° = 0

Putting values in the middle function:
⇒ 1 – 2×

⇒

⇒ 0

Putting values in the rightmost function:
⇒ 2×

⇒

⇒ 0

Therefore, all simplify to 0 and are equal.

Hence verified.

(v) We know sin 60° = (√3/2) (From the table)

cos 60° = (1/2)

tan 60° = √3

and sec 30° = 2

Putting values in the left hand side:
⇒

⇒

⇒

Putting values in the right hand side:
⇒

Therefore, left hand side and right hand side are equal.

Hence verified.

(vi) We know cos 60° = (1/2) (From the table)

tan 60° = √3

Putting values in the left hand side:
⇒

⇒

⇒

Putting values in the right hand side:
⇒

⇒

⇒

⇒

Therefore, left hand side and right hand side are equal.

Hence verified.

(vii) We know sec 60° = (2/√3) (From the table)

tan 30° = (1/√3)

sin 30° = (1/2)

Putting values in the left hand side:
⇒

⇒

⇒ 3

Putting values in the right hand side:
⇒

⇒

⇒ 3

Therefore, left hand side and right hand side are equal.

Hence verified.

(viii) We know sin 30° = (1/2) (From the table)

cot 30° = √3

cosec 45° = √2

tan 60° = √3

and tan 45° = 1

Putting the values in the left hand side:

⇒

⇒

⇒ 0

Which is equal to the right hand side.

Hence verified.

(ix) We know cos 60° = (1/2) (From the table)

sin 60° = (√3/2)

sec 60° = 2

and cot 45° = 1

Putting the values in the left hand side:

⇒

⇒

⇒ 1

Which is equal to the right hand side.

Hence verified.

(x) We know sin 30° = cos 60° = (1/2) (From the table)

Cos 30° = sin 60° = (√3/2)

Sin 90° = 1

So the right hand side = sin 90° = 1

Putting the values in the left hand side:

⇒

⇒

⇒ 1

Which is equal to the right hand side.

Hence verified.