CBSE Class 10 Mathematics Introduction to Trigonometry VBQs Set 08

Question. Prove that: \( \frac{1 + \sin \theta}{1 - \sin \theta} = (\sec \theta + \tan \theta)^2 \)
Answer: \( \frac{1 + \sin \theta}{1 - \sin \theta} = (\sec \theta + \tan \theta)^2 \)

Question. Prove that: \( \frac{\cot A + \tan B}{\cot B + \tan A} = \cot A \cdot \tan B \)
Answer: \( \frac{\cot A + \tan B}{\cot B + \tan A} = \cot A \cdot \tan B \)

Question. Prove that: \( \frac{\tan^2 \theta}{\tan^2 \theta - 1} + \frac{\csc^2 \theta}{\sec^2 \theta - \csc^2 \theta} = \frac{1}{\sin^2 \theta - \cos^2 \theta} \)
Answer: \( \frac{\tan^2 \theta}{\tan^2 \theta - 1} + \frac{\csc^2 \theta}{\sec^2 \theta - \csc^2 \theta} = \frac{1}{\sin^2 \theta - \cos^2 \theta} \)

Question. Prove that: \( \frac{\cos \theta}{\csc \theta + 1} + \frac{\cos \theta}{\csc \theta - 1} = 2 \tan \theta \)
Answer: \( \frac{\cos \theta}{\csc \theta + 1} + \frac{\cos \theta}{\csc \theta - 1} = 2 \tan \theta \)

Question. Prove that: \( 2 \sec^2 \theta - \sec^4 \theta - 2 \csc^2 \theta + \csc^4 \theta = \cot^4 \theta - \tan^4 \theta \)
Answer: \( 2 \sec^2 \theta - \sec^4 \theta - 2 \csc^2 \theta + \csc^4 \theta = \cot^4 \theta - \tan^4 \theta \)

Question. Prove that: \( \frac{\cos \theta}{1 - \sin \theta} + \frac{\cos \theta}{1 + \sin \theta} = 2 \sec \theta \)
Answer: \( \frac{\cos \theta}{1 - \sin \theta} + \frac{\cos \theta}{1 + \sin \theta} = 2 \sec \theta \)

Question. Prove that: \( \frac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta} = \tan \theta \)
Answer: \( \frac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta} = \tan \theta \)

Question. Prove that: \( \frac{1 + \cos \theta}{1 - \cos \theta} = (\csc \theta + \cot \theta)^2 \)
Answer: \( \frac{1 + \cos \theta}{1 - \cos \theta} = (\csc \theta + \cot \theta)^2 \)

Question. Prove that: \( \left( 1 + \frac{1}{\tan^2 \theta} \right) \left( 1 + \frac{1}{\cot^2 \theta} \right) = \frac{1}{\sin^2 \theta - \sin^4 \theta} \)
Answer: \( \left( 1 + \frac{1}{\tan^2 \theta} \right) \left( 1 + \frac{1}{\cot^2 \theta} \right) = \frac{1}{\sin^2 \theta - \sin^4 \theta} \)

Question. Prove that: \( \frac{1}{\sec \theta - \tan \theta} = \sec \theta + \tan \theta \)
Answer: \( \frac{1}{\sec \theta - \tan \theta} = \sec \theta + \tan \theta \)

Question. Prove that: \( \frac{1}{\sec x - \tan x} - \frac{1}{\cos x} = \frac{1}{\cos x} - \frac{1}{\sec x + \tan x} \)
Answer: \( \frac{1}{\sec x - \tan x} - \frac{1}{\cos x} = \frac{1}{\cos x} - \frac{1}{\sec x + \tan x} \)

Question. Prove that: \( \sqrt{\frac{\sec \theta - 1}{\sec \theta + 1}} + \sqrt{\frac{\sec \theta + 1}{\sec \theta - 1}} = 2 \csc \theta \)
Answer: \( \sqrt{\frac{\sec \theta - 1}{\sec \theta + 1}} + \sqrt{\frac{\sec \theta + 1}{\sec \theta - 1}} = 2 \csc \theta \)

Question. Prove that: \( \sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}} + \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} = 2 \csc \theta \)
Answer: \( \sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}} + \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} = 2 \csc \theta \)

Question. Prove that: \( \left( \tan \theta + \frac{1}{\cos \theta} \right)^2 + \left( \tan \theta - \frac{1}{\cos \theta} \right)^2 = 2 \left( \frac{1 + \sin^2 \theta}{1 - \sin^2 \theta} \right) \)
Answer: \( \left( \tan \theta + \frac{1}{\cos \theta} \right)^2 + \left( \tan \theta - \frac{1}{\cos \theta} \right)^2 = 2 \left( \frac{1 + \sin^2 \theta}{1 - \sin^2 \theta} \right) \)

Question. Prove that: \( \frac{\cos^2 \theta}{1 - \tan \theta} + \frac{\sin^3 \theta}{\sin \theta - \cos \theta} = 1 + \sin \theta \cos \theta \)
Answer: \( \frac{\cos^2 \theta}{1 - \tan \theta} + \frac{\sin^3 \theta}{\sin \theta - \cos \theta} = 1 + \sin \theta \cos \theta \)

Question. Prove that: \( \tan^2 A \sec^2 B - \sec^2 A \tan^2 B = \tan^2 A - \tan^2 B \)
Answer: \( \tan^2 A \sec^2 B - \sec^2 A \tan^2 B = \tan^2 A - \tan^2 B \)

Question. Prove that: \( \frac{\tan A + \tan B}{\cot A + \cot B} = \tan A \cdot \tan B \)
Answer: \( \frac{\tan A + \tan B}{\cot A + \cot B} = \tan A \cdot \tan B \)

Question. Prove that: \( \frac{(1 + \sin \theta)^2 + (1 - \sin \theta)^2}{2 \cos^2 \theta} = \frac{1 + \sin^2 \theta}{1 - \sin^2 \theta} \)
Answer: \( \frac{(1 + \sin \theta)^2 + (1 - \sin \theta)^2}{2 \cos^2 \theta} = \frac{1 + \sin^2 \theta}{1 - \sin^2 \theta} \)

Question. Prove that: \( \frac{\sin A + \cos A}{\sin A - \cos A} + \frac{\sin A - \cos A}{\sin A + \cos A} = \frac{2}{\sin^2 A - \cos^2 A} = \frac{2}{2 \sin^2 A - 1} \)
Answer: \( \frac{\sin A + \cos A}{\sin A - \cos A} + \frac{\sin A - \cos A}{\sin A + \cos A} = \frac{2}{\sin^2 A - \cos^2 A} = \frac{2}{2 \sin^2 A - 1} \)

Question. Prove that: \( \frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} + \frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta} = \frac{2}{1 - 2 \cos^2 \theta} \)
Answer: \( \frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} + \frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta} = \frac{2}{1 - 2 \cos^2 \theta} \)

Question. Prove that: \( \frac{\cot A + \csc A - 1}{\cot A - \csc A + 1} = \frac{1 + \cos A}{\sin A} \)
Answer: \( \frac{\cot A + \csc A - 1}{\cot A - \csc A + 1} = \frac{1 + \cos A}{\sin A} \)

Question. Prove that: \( \sin^4 A + \cos^4 A = 1 - 2 \sin^2 A \cos^2 A \)
Answer: \( \sin^4 A + \cos^4 A = 1 - 2 \sin^2 A \cos^2 A \)

Question. Prove that: \( (\sec A - \csc A)(1 + \tan A + \cot A) = \tan A \sec A - \cot A \csc A \)
Answer: \( (\sec A - \csc A)(1 + \tan A + \cot A) = \tan A \sec A - \cot A \csc A \)

Question. Prove that: \( \csc^6 \theta = \cot^6 \theta + 1 + 3 \cot^2 \theta \csc^2 \theta \)
Answer: \( \csc^6 \theta = \cot^6 \theta + 1 + 3 \cot^2 \theta \csc^2 \theta \)

Question. Prove that: \( (\tan A + \csc B)^2 - (\cot B - \sec A)^2 = 2 \tan A \cot B (\csc A + \sec B) \)
Answer: \( (\tan A + \csc B)^2 - (\cot B - \sec A)^2 = 2 \tan A \cot B (\csc A + \sec B) \)

Question. Prove that: \( \frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta \)
Answer: \( \frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta \)

Question. Prove that: \( \frac{\cos A}{1 - \sin A} + \frac{\sin A}{1 - \cos A} + 1 = \frac{\sin A \cos A}{(1 - \sin A)(1 - \cos A)} \)
Answer: \( \frac{\cos A}{1 - \sin A} + \frac{\sin A}{1 - \cos A} + 1 = \frac{\sin A \cos A}{(1 - \sin A)(1 - \cos A)} \)

Question. Prove that: \( \frac{1 - \cos \theta + \sin \theta}{1 + \cos \theta - \sin \theta} = \frac{1 + \sin \theta}{\cos \theta} \)
Answer: \( \frac{1 - \cos \theta + \sin \theta}{1 + \cos \theta - \sin \theta} = \frac{1 + \sin \theta}{\cos \theta} \)

Question. Prove that: \( \frac{\csc A}{\csc A - 1} + \frac{\csc A}{\csc A + 1} = 2 \sec^2 A \)
Answer: \( \frac{\csc A}{\csc A - 1} + \frac{\csc A}{\csc A + 1} = 2 \sec^2 A \)

Question. Prove that: \( \frac{\cos A}{1 - \tan A} + \frac{\sin A}{1 - \cot A} = \sin A + \cos A \)
Answer: \( \frac{\cos A}{1 - \tan A} + \frac{\sin A}{1 - \cot A} = \sin A + \cos A \)

Question. Prove that: \( \frac{\tan^3 \theta}{1 + \tan^2 \theta} + \frac{\cot^3 \theta}{1 + \cot^2 \theta} = \sec \theta \csc \theta - 2 \sin \theta \cos \theta \)
Answer: \( \frac{\tan^3 \theta}{1 + \tan^2 \theta} + \frac{\cot^3 \theta}{1 + \cot^2 \theta} = \sec \theta \csc \theta - 2 \sin \theta \cos \theta \)

Question. Prove that: \( (1 + \cot A + \tan A)(\sin A - \cos A) = \frac{\sec A}{\csc^2 A} - \frac{\csc A}{\sec^2 A} \)
Answer: \( (1 + \cot A + \tan A)(\sin A - \cos A) = \frac{\sec A}{\csc^2 A} - \frac{\csc A}{\sec^2 A} \)

Question. Prove that: \( \frac{\cot^2 A (\sec A - 1)}{1 + \sin A} = \sec^2 A \left( \frac{1 - \sin A}{1 + \sec A} \right) \)
Answer: \( \frac{\cot^2 A (\sec A - 1)}{1 + \sin A} = \sec^2 A \left( \frac{1 - \sin A}{1 + \sec A} \right) \)

Question. Prove that: \( \frac{\tan A}{(1 + \tan^2 A)^2} + \frac{\cot A}{(1 + \cot^2 A)^2} = \sin A \cos A \)
Answer: \( \frac{\tan A}{(1 + \tan^2 A)^2} + \frac{\cot A}{(1 + \cot^2 A)^2} = \sin A \cos A \)