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 tan^{2} A = 1 – e^{2} prove that, sec A + tan^{3} A csc A = (2 – e^{2})^{3/2}.
5. If tan β + cot β = 2, prove that tan^{3} β + cot^{3} β =2.
6. If cos θ + sec θ = 2, prove that cos^{4} θ + sec^{4} θ =2.
Hint: cos^{2 }θ - 2 cos θ + 1 = 0
⟹ (cos θ - 1)^{2} = 0
⟹ cos θ - 1 = 0
⟹ cos θ = 1
⟹ sec θ = 1
7. If tan^{2} A = 1 + 2 tan^{2} B, prove that cos^{2} B = 2 cos^{2} A
Hint: tan^{2} A = 1 + 2 tan^{2} B
⟹ sec^{2} A - 1 = 1 + 2 (sec^{2} B - 1)
⟹ sec^{2} A - 1 = 1 + 2 sec^{2} B - 2
⟹ sec^{2} A - 1 = 2 sec^{2} B - 1
8. If cos A + sec A = \(\sqrt{3}\) show that, cos^{3} A + sec^{3} A = 0.
9. If cos^{2} A – sin^{2} A = tan^{2} B, prove that tan^{2} A = cos^{2} B – sin^{2} B.
Hint: cos^{2} A – sin^{2} A = tan^{2} B
⟹ cos^{2} A – (1 - cos^{2} A) = sec^{2} B - 1
⟹ cos^{2} A – 1 + cos^{2} A = sec^{2} B - 1
⟹ 2 cos^{2} A – 1 = sec^{2} B - 1
⟹ 2 cos^{2} A = sec^{2} B
⟹ 2 \(\frac{1}{sec^{2} A}\) = \(\frac{1}{cos^{2} B}\)
⟹ sec^{2} A = 2 cos^{2} B
⟹ 1 + tan^{2} A = cos^{2} B + cos^{2} B
⟹ tan^{2} A = cos^{2} B + cos^{2} B - 1
⟹ tan^{2} A = cos^{2} B - 1 + cos^{2} B
⟹ tan^{2} A = cos^{2} B - (1 - cos^{2} B)
10. If a^{2} sec^{2} θ – b^{2} tan^{2} θ = c^{2}, 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 + 4x^{2}, prove that, sec β + tan β = 2x or, \(\frac{1}{2x}\).
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