Prove the following sin θ 1 + cos θ + 1 + cos θ sin θ = 2 cosec θ

L H S = sin θ 1 + cos θ + 1 + cos θ sin θ = sin 2 θ + 1 + cos θ 2 sin θ 1 + cos θ = sin 2 θ + 1 + cos 2 θ + 2 cos θ sin θ 1 + cos θ ∵ a + b 2 = a 2 + b 2 + 2 a b = 1 + 1 + 2 cos θ sin θ 1 + cos θ ∵ sin 2 θ + cos 2 θ = 1 = 2 1 + cos θ sin θ 1 + cos θ = 2 sin θ = 2 c osec θ = R H S ∵ cosec θ = 1 sin θ

Prove the following tan A 1 + sec A - tan A 1 - sec A = 2 cosec A

L H S = tan A 1 + sec A - tan A 1 - sec A = tan A 1 - sec A ) - tan ( 1 + sec A 1 + sec A 1 - sec A = tan A - 2 sec A 1 - sec 2 A = 2 tan A · sec A se c 2 A - 1 ∵ a + b a - b = a 2 - b 2 = 2 tan A · sec A tan 2 A ∵ sec 2 A - tan 2 A = 1 ∵ sec θ = 1 cos θ and tan θ = sin θ cos θ = 2 sec A tan A = 2 sin A = 2 cosec A = RHS ∵ cosec θ = 1 sin θ

Prove the following : If tan A = 3 4 , then sin A · cos A = 12 25 .

Given, tan A = 3 4 = P B = Perpendicular Base Let P = 3 k and B = 4 k By Pythagoras theorem, H 2 = P 2 + B 2 = 3 k 2 + 4 k 2 = 9 k 2 + 16 k 2 = 25 k 2 ⇒ H = 5 k [since, side cannot be negative] ∴ sin A = P H = 3 k 5 k = 3 5 and cos A = B H = 4 k 5 k = 4 5 Now, sin A cos A = 3 5 · 4 5 = 12 25

E M B I B E

NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS>Chapter 8 - Introduction to Trigonometry and its Applications>Exercise>Q 7

HARD

10th CBSE

IMPORTANT

Earn 100

If the length of the shadow of a tower is increasing, then the angle of elevation of the Sun is also increasing.

66.67% studentsanswered this correctly

Important Questions on Introduction to Trigonometry and its Applications

HARD

10th CBSE

IMPORTANT

NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS>Chapter 8 - Introduction to Trigonometry and its Applications>Exercise>Q 8

If a man standing on a platform

3 m

above the surface of a lake observes a cloud and its reflection in the lake, then the angle of elevation of the cloud is equal to the angle of depression of its reflection.