# PRACTICE PROBLEMS IN TRIGONOMETRY FOR GRADE 10

Practice Problems in Trigonometry for Grade 10 :

Here we are going to see some practice problems in the topic trigonometry.

## Practice Problems in Trigonometry for Grade 10 - Quetsions

(1)  Determine whether each of the following is an identity or not.

(i) cos2θ + sec2θ  =  2 + sin θ

(ii) cot2θ + cos θ  =  sin2θ

(2)  Prove the following identities

(i) sec2θ + cosec2θ  =  sec2θ ⋅ cosec2θ

Question 1 :

Determine whether each of the following is an identity or not.

(i) cos2θ + sec2θ  =  2 + sinθ

Solution :

To check whether the given question is an identity or not,first  we have to simplify the left side question. So that we have to get the right side answer.

L.H.S

=  cos2θ + sec2θ

=  1 - sin2θ + 1 + tan2θ

=  2 + tan2θ - sin2θ

Since it is not equal to R.H.S, it is not an identity.

(ii) cot2θ + cos θ  =  sin2θ

Solution :

L.H.S

=  cot2θ + cosθ

=  cosec2θ - 1 + cosθ

Since it is not equal to R.H.S, it is not an identity.

Question 2 :

Prove the following identities

(i)  sec2θ + cosec2θ  =  sec2θ ⋅ cosec2θ

Solution :

L.H.S

=  sec2θ + cosec2θ

=  (1/cos2θ) + (1/sin2θ)

=  (sin2θ + cos2θ)/(cos2θ ⋅ sin2θ)

Value of sin2θ + cos2θ  =  1.

Then,

=  1/(cos2θ sin2θ)

=  (1/cos2θ)(1/sin2θ)

=  sec2θ  cosec2θ

(ii)  sin θ /(1-cos θ) = cosec θ + cot θ

Solution :

L.H.S

=  sin θ /(1-cos θ)

Multiplying by the conjugate of denominator we get,

=  [sin θ /(1-cos θ)]  [(1+cos θ)/(1+cos θ)]

instead of (1 + cos θ)/(1 + cos θ) we can write 1 - cos²θ by using the algebraic formula.

=  [sin θ (1 + cosθ)]/(1-cos2θ)

=  [sin θ (1 + cosθ)]/sin2θ

=  (1 + cosθ)]/sinθ

=  (1/sinθ) + (cosθ/sinθ)

=  cosec θ + cot θ

(iii)  √(1 - sin θ)/(1 + sin θ) = sec θ - tan θ

Solution :

L.H.S

=  √(1-sin θ)/(1+sin θ)

=  (1-sin θ)/(1+sin θ)  (1-sin θ)/(1-sin θ)

=  (1-sin θ)2/[(1+sin θ) x (1-sin θ)]

=  (1-sin θ)2/(12sin2θ)

=  (1-sin θ)2/(cos2θ)

=  [(1-sin θ)/(cosθ)]2

=  [(1-sin θ)/(cosθ)]

=  [(1/cosθ)-(sin θ/cosθ)]

=  sec θ - tan θ

R.H.S

After having gone through the stuff above, we hope that students would have understood how to solve word problems in trigonometry.

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