CBSE Class 10 Mathematics Standard 2023: question 30

Hints

Open one hint at a time and try again before the next.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

1. Write everything in sin and cos:
   LHS $=\frac{\frac{\sin \theta }{\cos \theta }}{1-\frac{\cos \theta }{\sin \theta }}+\frac{\frac{\cos \theta }{\sin \theta }}{1-\frac{\sin \theta }{\cos \theta }}$

   ½ mark

2. Simplify each fraction:
   $=\frac{{\sin }^2\theta }{\cos \theta (\sin \theta -\cos \theta )}-\frac{{\cos }^2\theta }{\sin \theta (\sin \theta -\cos \theta )}$

   1 mark

3. Take a common denominator:
   $=\frac{1}{\sin \theta -\cos \theta }\left[\frac{{\sin }^3\theta -{\cos }^3\theta }{\sin \theta \cos \theta }\right]$

   ½ mark

4. Factorise the difference of cubes:
   $=\frac{(\sin \theta -\cos \theta )({\sin }^2\theta +\sin \theta \cos \theta +{\cos }^2\theta )}{(\sin \theta -\cos \theta )\sin \theta \cos \theta }$

   ½ mark

5. Use ${\sin }^2\theta +{\cos }^2\theta =1$:
   $=\frac{1+\sin \theta \cos \theta }{\sin \theta \cos \theta }=\frac{1}{\sin \theta \cos \theta }+1=1+\sec \theta \mathrm{cosec}\theta =$ RHS

   ½ mark

Answer: LHS = 1 + sec θ cosec θ = RHS (proved)

Revise the chapter

• Chapter: Introduction to Trigonometry (notes, formulas, where marks are lost and every past question)
• Topic: Proving Simple Trigonometric Identities