CBSE Class 10 Mathematics Standard 2024: question 28

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 sines and cosines:

   LHS $=\frac{\frac{\sin \theta }{\cos \theta }}{\frac{\sin \theta -\cos \theta }{\sin \theta }}+\frac{\frac{\cos \theta }{\sin \theta }}{\frac{\cos \theta -\sin \theta }{\cos \theta }}$

   ½ mark

2. Simplify each fraction:

   $=\frac{{\sin }^2\theta }{\cos \theta (\sin \theta -\cos \theta )}-\frac{{\cos }^2\theta }{\sin \theta (\sin \theta -\cos \theta )}=\frac{1}{\sin \theta -\cos \theta }\left[\frac{{\sin }^2\theta }{\cos \theta }-\frac{{\cos }^2\theta }{\sin \theta }\right]$

   1 mark

3. Take the LCM and use $a^3-b^3=(a-b)(a^2+ab+b^2)$:

   $=\frac{1}{\sin \theta -\cos \theta }\cdot \frac{{\sin }^3\theta -{\cos }^3\theta }{\sin \theta \cos \theta }=\frac{(\sin \theta -\cos \theta )({\sin }^2\theta +{\cos }^2\theta +\sin \theta \cos \theta )}{(\sin \theta -\cos \theta )\sin \theta \cos \theta }$

   1 mark

4. Cancel $(\sin \theta -\cos \theta )$ and 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. Hence proved.

   ½ mark

Answer: LHS simplifies to $1+\sec \theta \mathrm{cosec}\theta$ = RHS. Hence proved.

Revise the chapter

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