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 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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10th Grade Math
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