# PRACTICE PROBLEMS IN TRIGONOMETRY FOR GRADE 10

Problem 1 :

Determine whether the following is an identity or not.

cos2θ + sec2θ  =  2 + sinθ

Solution :

cos2θ + sec2θ :

=  1 - sin2θ + 1 + tan2θ

=  2 + tan2θ - sin2θ

≠  2 + sinθ

Because cos2θ + sec2θ    2 + sinθ, it's not an identity.

Problem 2 :

Determine whether the following is an identity or not.

cot2θ + cos θ  =  sin2θ

Solution :

cot2θ + cosθ :

=  cosec2θ - 1 + cosθ

≠  sin2θ

Because cot2θ + cos θ    sin2θ, it's not an identity.

Problem 3 :

Prove the following identity.

sec2θ + cosec2θ  =  sec2θ ⋅ cosec2θ

Solution :

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

Problem 4 :

Prove the following identity.

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

Solution :

sin θ /(1-cos θ) :

Multiply both numerator and denominator by the conjugate of the denominator.

=  [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

Problem 5 :

Prove the following identity.

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

Solution :

(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

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