Tricky Trigonometry questions for IITJEE mains maths practice

Prove the following:

  1. \frac{\sin{A}}{1-\cos{A}} = \frac{1+\cos{A}-\sin{A}}{\sin{A}-1+\cos{A}}
  2. \frac{1+\sin{A}}{\cos{A}}=\frac{1+\sin{A}+\cos{A}}{\cos{A}+1-\sin{A}}
  3. \frac{\tan{A}}{\sec{A}-1}=\frac{\tan{A}+\sec{A}+1}{\sec{A}-1+\tan{A}}
  4. \frac{1+\csc{A}+ \cot{A}}{1+\csc{A}-\cot{A}} = \frac{\csc{A}+\cot{A}-1}{\cot{A}-\csc{A}+1}

Hint: Directly trying to prove LHS is RHS is difficult in all the above; or even trying to transform RHS to LHS is equally difficult; it is quite easier to prove the equivalent statement by taking cross-multiplication of the appropriate expressions. 🙂

Regards,

Nalin Pithwa.

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