Worksheet on Establishing Conditional Results Using Trigonometric Identities

In worksheet on establishing conditional results using Trigonometric identities we will prove various types of practice questions on Trigonometric identities.

Here you will get 12 different types of establishing conditional results using Trigonometric identities questions with some selected questions hints. 

1. If sin A + cos A = 1, prove that sin A - cos A = ± 1.

2. If csc θ + cot θ = a, prove that, cos θ = \(\frac{a^{2} - 1}{ a^{2} + 1}\).

3. If x cos θ + y sin θ = z, prove that

                     a sin θ + b cos θ = ± \(\sqrt{x^{2} + y^{2} + z^{2} }\).

4. If tan² A = 1 – e² prove that, sec A + tan³ A csc A = (2 – e²)^3/2.

5. If tan β + cot β = 2, prove that tan³ β + cot³ β =2.

6. If cos θ + sec θ = 2, prove that cos⁴ θ + sec⁴ θ =2.

Hint: cos² θ - 2 cos θ + 1 = 0

     ⟹ (cos θ - 1)² = 0

     ⟹ cos θ - 1 = 0

     ⟹ cos θ = 1

     ⟹ sec θ = 1

7. If tan² A = 1 + 2 tan² B, prove that cos² B = 2 cos² A

Hint: tan² A = 1 + 2 tan² B

     ⟹  sec² A - 1 = 1 + 2 (sec² B - 1)

     ⟹  sec² A - 1 = 1 + 2 sec² B - 2

     ⟹  sec² A - 1 = 2 sec² B - 1

8. If cos A + sec A = \(\sqrt{3}\) show that, cos³ A + sec³ A = 0.

9. If cos² A – sin² A = tan² B, prove that tan² A = cos² B – sin² B.

Hint: cos² A – sin² A = tan² B

     ⟹  cos² A – (1 - cos² A) = sec² B - 1

     ⟹  cos² A – 1 + cos² A = sec² B - 1

     ⟹  2 cos² A – 1 = sec² B - 1

     ⟹  2 cos² A = sec² B 

     ⟹  2 \(\frac{1}{sec^{2} A}\) = \(\frac{1}{cos^{2} B}\) 

     ⟹  sec² A = 2 cos² B 

     ⟹  1 + tan² A = cos² B + cos² B 

     ⟹  tan² A = cos² B + cos² B - 1

     ⟹  tan² A = cos² B - 1 + cos² B

     ⟹  tan² A = cos² B - (1 - cos² B)

10. If a² sec²  θ – b² tan² θ = c², show that sin θ = ±\(\sqrt{\frac{c^{2} – a^{2}}{c^{2} – b^{2}}}\).

11. If (1 – cos A)(1 – cos B)(1 – cos C) = (1 + cos A)(1 + cos B)(1 + cos C) then prove that each side is equal to ± sin A sin B sin C.

12. If 4x sec β = 1 + 4x², prove that, sec β + tan β = 2x or, \(\frac{1}{2x}\).

10th Grade Math

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