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1 mark

If cscA−cotA=x, then the value of x is:

  1. A
    \frac{\sin A}{1-\cos A}
  2. B
    \frac{\sin A}{1+\cos A}
  3. C
    \frac{\sin A}{1+\cos A}
  4. D
    \frac{\sin A}{1-\cos A}

Solution & Step-by-step Explanation

Given expression:
x=cscA−cotA
Expressing cscA and cotA in terms of sinA and cosA:

cscA=
sinA
1
​

cotA=
sinA
cosA
​

Substitute these values:

x=
sinA
1
​
−
sinA
cosA
​

x=
sinA
1−cosA
​

To match the options, multiply the numerator and denominator by (1+cosA):

x=
sinA(1+cosA)
(1−cosA)(1+cosA)
​

Using the algebraic identity (a−b)(a+b)=a
2
−b
2
:

x=
sinA(1+cosA)
1−cos
2
A
​

Since 1−cos
2
A=sin
2
A:

x=
sinA(1+cosA)
sin
2
A
​

x=
1+cosA
sinA
​

Practice this question

Try it yourself before checking the explanation above.

If cscA−cotA=x, then the value of x is:
A
\frac{\sin A}{1-\cos A}
B
\frac{\sin A}{1+\cos A}
C
\frac{\sin A}{1+\cos A}
D
\frac{\sin A}{1-\cos A}

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TrigonometryTrigonometric Identities

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