Question

Prove that:
$\frac{1 + cosec A}{cosec A} = \frac{cos ^{2} A}{1 - sin A}$

Approach

Simplify LHS: $\frac{1 + c o s A}{c o s A} = \frac{s i n A + s i n A c o s A}{c o s A} \div \frac{s i n A}{c o s A} = 1 + sin A$ . Then $(1 + sin A) = \frac{( 1 + s i n A ) ( 1 - s i n A )}{1 - s i n A} = \frac{1 - s i n ^{2} A}{1 - s i n A} = \frac{c o s ^{2} A}{1 - s i n A}$ , which equals RHS.

Step-by-step working

1
Rewrite LHS: $\frac{1 + \frac{1}{s i n A}}{\frac{1}{s i n A}} = \frac{s i n A + 1}{1} = 1 + sin A$ .
0.5 marks
2
Simplify: $1 + sin A = \frac{( 1 + s i n A ) ( 1 - s i n A )}{1 - s i n A} = \frac{1 - s i n ^{2} A}{1 - s i n A}$ .
1 mark
3
Use $1 - sin^{2} A = cos^{2} A$ : $\frac{c o s ^{2} A}{1 - s i n A}$ , which is RHS.
0.5 marks
4
Hence LHS $=$ RHS.
1 mark

Concepts used

trigonometric identityprovecoseccossin

Approach

Step-by-step working

Concepts used

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