Breath Math

1. Express the trigonometric ratios sin A, sec A and tan A in terms of cot A. 

Solution:

We know that,

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2. Write all the other trigonometric ratios of ∠A in terms of sec A. 

Solution:

We know that

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3. Evaluate (i) (sin²63° + sin²27°)/(cos²17°+cos²73°)

(ii) sin25° cos65° + cos25° sin65° 

Solution:

(i)

(ii)sin25° cos65° + cos25° sin65° 

= (sin25°){cos(90°-25°)} + cos25°{sin(90°-25°)}

= (sin25°)(sin25°)+(cos25°)(cos25°)

= (sin²25°) + cos²25°

= 1

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4. Choose the correct option. Justify your choice.

(i) 9 sec²A − 9 tan² A =

(A) 1

(B) 9

(C) 8

(D) 0

(ii) (1 + tan θ + sec θ) (1 + cot θ − cosec θ)

(A) 0

(B)1

(C) 2

(D) −1

(iii) (secA + tanA) (1 − sinA) =

(A) secA

(B) sinA

(C) cosecA

(D) cosA 

(iv)(1+tan²A)/(1+cot²A) =

(A) sec²A

(B) -1

© cot²A

(d) tan²A

Solution:

(i) 9 sec² A − 9 tan² A

= 9 (sec² A – tan² A)

= 9 (1)  [Because  sec2 A − tan2 A = 1]

= 9

(B) is correct.

 

(ii) (1 + tan θ + sec θ) (1 + cot θ − cosec θ)

(C) is correct.

 

(iii)

= cosA

(D) is correct.

(iv)

(D) is correct.

 

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5. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. 

Solution:

Solution:

(iii)

Solution:

= secθ cosec θ + 1 = R.H.S.

 

(iv)

Solution:

 

(v)

solution:

= cosec A + cot A = R.H.S

(vi)

Solution:

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