✓ Board-verified
Mathematics · 2025 · 2 marks
CBSE 2025 · Region 6 · Set 1 · Q22
(a) If a sec θ+ b tan θ= m and b sec θ+ a tan θ= n, prove that a^2+ n^2= b^2+ m^2
(a)
If a $\displaystyle \sec \theta+\mathrm{b} \tan \theta=\mathrm{m}$ and $\displaystyle \mathrm{b} \sec \theta+\mathrm{a} \tan \theta=\mathrm{n}$, prove that $\displaystyle \mathrm{a}^{2}+\mathrm{n}^{2}=\mathrm{b}^{2}+\mathrm{m}^{2}$
OR
(b)
Use the identity : $\displaystyle \sin ^{2} \mathrm{~A}+\cos ^{2} \mathrm{~A}=1$ to prove that $\displaystyle \tan ^{2} \mathrm{~A}+1=\sec ^{2} \mathrm{~A}$. Hence, find the value of $\displaystyle \tan \mathrm{A}$, when $\displaystyle \sec \mathrm{A}=\frac{5}{3}$, where A is an acute angle.
Marking-scheme solution
\[m^{2}=a^{2} \sec ^{2} \theta+b^{2} \tan ^{2} \theta+2 a b \sec \theta \tan \theta
\]
\(\displaystyle \mathrm{n}^{2}=\mathrm{b}^{2} \sec ^{2} \theta+\mathrm{a}^{2} \tan ^{2} \theta+2 \mathrm{ab} \sec \theta \tan \theta\)
\[m^{2}-n^{2}=a^{2}\left(\sec ^{2} \theta-\tan ^{2} \theta\right)+b^{2}\left(\tan ^{2} \theta-\sec ^{2} \theta\right)
\]
\(\displaystyle \Rightarrow \mathrm{m}^{2}-\mathrm{n}^{2}=\mathrm{a}^{2}-\mathrm{b}^{2}\) or \(\displaystyle \mathrm{a}^{2}+\mathrm{n}^{2}=\mathrm{m}^{2}+\mathrm{b}^{2}\)
\(\displaystyle \sin ^{2} \mathrm{~A}+\cos ^{2} \mathrm{~A}=1\)
Dividing both sides by \(\displaystyle \cos ^{2} \mathrm{~A}\), we get
\[\frac{\sin ^{2} A}{\cos ^{2} A}+\frac{\cos ^{2} A}{\cos ^{2} A}=\frac{1}{\cos ^{2} A}
\]
\(\displaystyle \tan ^{2} \mathrm{~A}+1=\sec ^{2} \mathrm{~A}\)
\(\displaystyle \tan ^{2} \mathrm{~A}+1=\left(\frac{5}{3}\right)^{2}\)
\(\displaystyle \tan \mathrm{A}=\frac{4}{3}\)
All Introduction to Trigonometry questions2025 question paperNCERT solutions: Introduction to TrigonometryPractice Introduction to Trigonometry →
More from Introduction to Trigonometry
- Prove that: (1+ cot θ-cosec θ)(1+ tan θ+ sec θ)=22026
- Evaluate: (5 cos^2 60^°+4 sec^2 30^°- tan^2 45^°)/(sin^2 30^°+ cos^2 30^°) OR Prove that: 1+(cot^2 α)/(1+cosec α)=cosec α2026
- [5/8 sec^2 60^°- tan^2 60^°+ cos^2 45^°] is equal to2023
- If sin θ+ cos θ=√3, then prove that tan θ+ cot θ=1 OR Prove that: ( sin A+ sec A)^2+( cos A+cosec A)^2=(1+ sec A cosec A)^22026
- The hour-hand of a clock is 6 cm long. The angle swept by it between 7:20 a.m. and 7:55 a.m. is:2023
- If x=a cos θ and y=b sin θ, then the value of b^2 x^2+a^2 y^2 is:2024
- Assertion: For an acute angle θ, value of cosec θ cannot be 1/(√2). Reason (R): cosec θ ≥ 1 for 0^° ≤ θ ≤ 90^°2025
- Prove that (cosec A- sin A)( sec A- cos A)=1/(cot A+ tan A).2023
CBSE Class 10 Mathematics past-paper question from the 2025 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.