Question

From the adjacent figure 1) prove that sin 2a + cos² a = sin² theta+ cos² theta.
2) prove that sec²a - tan² a = sec²theta - tan² theta







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Answer 1

EXPLANATION.

Prove that.

(1) sin²α + cos²α = sin²θ + cos²θ.

(2) sec²α - tan²α = sec²θ - tan²θ.

As we know that,

Concept of Pythagoras Theorem.

(Hypotenuse)² = (Perpendicular)² + (Base)².

⇒ H² = P² + B².

Hypotenuse > Perpendicular > Base.

Using this concept in the equation, we get.

It is given that,

For θ angles.

Hypotenuse = 13 cm.

Base = 5 cm.

⇒ H² = P² + B².

⇒ (13)² = (P)² + (5)².

⇒ 169 = P² + 25.

⇒ 169 - 25 = P².

⇒ 144 = P².

⇒ P = √144.

⇒ P = 12 cm.

We can write as,

⇒ sinθ = p/h = 12/13.

⇒ cosθ = b/h = 5/13.

⇒ tanθ = 12/5.

For α angles.

Hypotenuse = 13 cm.

Perpendicular = 5 cm.

⇒ H² = P² + B².

⇒ (13)² = (5)² + B².

⇒ 169 - 25 = B².

⇒ 144 = B².

⇒ B = √144.

⇒ B = 12 cm.

We can write as,

⇒ sinα = p/h = 5/13.

⇒ cosα = b/h = 12/13.

⇒ tanα = p/b = 5/12.

Prove that.

(1) sin²α + cos²α = sin²θ + cos²θ.

Put the values in the equation, we get.

⇒ (5/13)² + (12/13)² = (12/13)² + (5/13)².

⇒ (25/169) + (144/169) = (144/169) + (25/169).

⇒ (169/169) = (169/169).

⇒ 1 = 1.

(2) sec²α - tan²α = sec²θ - tan²θ.

⇒ (13/12)² - (5/12)² = (13/5)² - (12/5)².

⇒ (169/144) - (25/144) = (169/25) - (144/25).

⇒ (144/144) = (25/25).

⇒ 1 = 1.

Hence Proved.

Answer 2

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EXPLANATION.

Prove that.

(1) sin³a + cos²a = sin²e + cos²0.

(2) sec²a-tan'a = sec²0-tan²0.

As we know that.

Concept of Pythagoras Theorem.

(Hypotenuse) (Perpendicular)² +

(Base).

- H²= P² + B².

Hypotenuse > Perpendicular > Base.

Using this concept in the equation, we

get.

It is given that.

For 8 angles.

Hypotenuse = 13 cm.

Base = 5 cm.

H² = P² + B²

(13)² = (P)² + (5)².

→ 169 = P² + 25.

169-25=p².

- 144 - p²

P= √144.

→ P = 12 cm.

We can write as,

→ sine=p/h = 12/13

cose = b/h = 5/13.

- tane = 12/5.

For a angles.

Hypotenuse = 13 cm.

Perpendicular = 5 cm.

→ H² = P² + B².

(13)² = (5)² + B².

-169-25-B².

- 144=B².

-B=√144.

B = 12 cm.

We can write as,

→ sina = p/h = 5/13.

→ cosa = b/h = 12/13.

→ tana = p/b = 5/12.

Prove that.

(1) sin²a + cos²a = sin²0 + cos²0.

Put the values in the equation, we get.

→ (5/13)² + (12/13)² = (12/13)² + (5/13)².

(25/169) + (144/169) = (144/169) +

(25/169).

(169/169) = (169/169).

⇒ 1 = 1.

(2) sec²a - tan²a = sec²0 - tan²0.

(13/12)²-(5/12)² = (13/5)² - (12/5)².

(169/144) - (25/144) = (169/25) - (144/25).

(144/144) = (25/25).

→ 1=1.

Hence Proved.