CBSE Class 10 Mathematics Standard 2023: question 30

Hints

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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. Replace 1 in the numerator by ${\sec }^2\theta -{\tan }^2\theta$ [since ${\sec }^2\theta -{\tan }^2\theta =1$]:

   LHS $=\frac{(\tan \theta +\sec \theta )-({\sec }^2\theta -{\tan }^2\theta )}{\tan \theta -\sec \theta +1}$

   1 mark

2. Factorise ${\sec }^2\theta -{\tan }^2\theta =(\sec \theta +\tan \theta )(\sec \theta -\tan \theta )$:

   $=\frac{(\tan \theta +\sec \theta )[1-\sec \theta +\tan \theta ]}{\tan \theta -\sec \theta +1}$

   1 mark

3. The bracket $1-\sec \theta +\tan \theta$ equals the denominator, so it cancels:

   $=\tan \theta +\sec \theta$

   ½ mark

4. $=\frac{\sin \theta }{\cos \theta }+\frac{1}{\cos \theta }=\frac{1+\sin \theta }{\cos \theta }=$ RHS. Hence proved.

   ½ mark

Answer: LHS = $\tan \theta +\sec \theta =\frac{1+\sin \theta }{\cos \theta }$ = RHS (proved)

Revise the chapter

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