QQuestion
Factorize: sin³ θ + cos³ θ.
Hence, prove the identity:
\frac{\sin^3\theta+\cos^3\theta}{\sin\theta+\cos\theta}+\sin\theta\cos\theta=1✓Answer
Solution:
Using a³ + b³ = (a + b)(a² − ab + b²):
sin³ θ + cos³ θ
= (sin θ + cos θ)(sin² θ − sin θ cos θ + cos² θ)
Since sin² θ + cos² θ = 1:
sin³ θ + cos³ θ = (sin θ + cos θ)(1 − sin θ cos θ) … (1)
Now consider the left-hand side of the identity:
\frac{\sin^3\theta+\cos^3\theta}{\sin\theta+\cos\theta}+\sin\theta\cos\thetaSubstituting from (1):
\frac{(\sin\theta+\cos\theta)(1-\sin\theta\cos\theta)}{\sin\theta+\cos\theta}+\sin\theta\cos\theta= 1 − sin θ cos θ + sin θ cos θ
= 1
Therefore, LHS = RHS. Hence proved.
Related Questions
More ICSE Moderate level questions
Statement (i): sin² θ + cos² θ = 1 Statement (ii): cosec² θ + cot² θ = 1 Which of the following is valid? (a) Only (i) (b) Only (ii) (c) Both (i) and (ii) (d) Neither (i) nor (ii)
Assertion (A): If sin² A + sin A = 1, then cos⁴ A + cos² A = 1. Reason (R): 1 − sin² A = cos² A. (a) A is true, R is false. (b) A is false, R is true. (c) Both A and R are true, and R is the correct reason for A. (d) Both A and R are true, and R is not the correct reason for A.
Prove the following identity: (sin2 θ − 1)(tan2 θ + 1) + 1 = 0
(1 + sin A)(1 − sin A) is equal to: (a) cosec2 A (b) sin2 A (c) sec2 A (d) cos2 A
Prove that:
Mr Gupta invested ₹33,000 in buying ₹100 shares of a company at 10% premium. The dividend declared by the company is 12%. Find: (a) the number of shares purchased by him. (b) his annual dividend.
The letters A, D, M, N, O, S, U and Y of the English alphabet are written on separate cards and put in a box. The cards are well shuffled and one card is drawn at random. What is the probability that the card drawn is a letter: (a) of the word MONDAY? (b) which does not appear in MONDAY? (c) which appears both in SUNDAY and MONDAY?
The nth term of an Arithmetic Progression (A.P.) is given by the relation Tn = 6(7 − n). Find: (a) its first term and common difference. (b) the sum of its first 25 terms.
The following table gives the duration of movies in minutes: Duration (minutes) Number of movies 100–110 5 110–120 10 120–130 17 130–140 8 140–150 6 150–160 4 Using the step-deviation method, find the mean duration of the movies.
(a) If the lines kx − y + 4 = 0 and 2y = 6x + 7 are perpendicular to each other, find the value of k. (b) Find the equation of a line parallel to 2y = 6x + 7 and passing through (−1, 1).
The polynomial 3x³ + 8x² − 15x + k has (x − 1) as a factor. Find the value of k. Hence, factorize the resulting polynomial completely.
Rohan bought the following eatables for his friends: S. No. Item Price Quantity Rate of GST 1 Laddu ₹500 per kg 2 kg 5% 2 Pastries ₹100 per piece 12 pieces 18% Calculate: (a) the total GST paid. (b) the total bill amount including GST.