Photo AI

Simplify the following to a single trigonometric ratio: 4.1 cos θ (tan θ + cot θ) 4.2 sin²(180° + B) · cosec(π - B) sec(2π - B) · cos(180° - B) - NSC Technical Mathematics - Question 4 - 2021 - Paper 2

Question icon

Question 4

Simplify-the-following-to-a-single-trigonometric-ratio:--4.1-cos-θ-(tan-θ-+-cot-θ)--4.2-sin²(180°-+-B)-·-cosec(π---B)-sec(2π---B)-·-cos(180°---B)-NSC Technical Mathematics-Question 4-2021-Paper 2.png

Simplify the following to a single trigonometric ratio: 4.1 cos θ (tan θ + cot θ) 4.2 sin²(180° + B) · cosec(π - B) sec(2π - B) · cos(180° - B)

Worked Solution & Example Answer:Simplify the following to a single trigonometric ratio: 4.1 cos θ (tan θ + cot θ) 4.2 sin²(180° + B) · cosec(π - B) sec(2π - B) · cos(180° - B) - NSC Technical Mathematics - Question 4 - 2021 - Paper 2

Step 1

4.1 cos θ (tan θ + cot θ)

96%

114 rated

Answer

We start by rewriting the expression:

extcosθ(tanθ+cotθ)=extcosθ(sin⁡θcos⁡θ+cos⁡θsin⁡θ)=cos θ(sin⁡2θ+cos⁡2θsin⁡θcos⁡θ) ext{cos } θ (tan θ + cot θ) = ext{cos } θ \left( \frac{\sin θ}{\cos θ} + \frac{\cos θ}{\sin θ} \right) = \text{cos } θ \left( \frac{\sin^2 θ + \cos^2 θ}{\sin θ \cos θ} \right)

Using the Pythagorean identity, (\sin^2 θ + \cos^2 θ = 1), we simplify:

=cos θ(1sin⁡θcos⁡θ)=cos θsin⁡θcos θ=1sin⁡θ=csc⁡θ.= \text{cos } θ \left( \frac{1}{\sin θ \cos θ} \right) = \frac{\text{cos } θ}{\sin θ \text{cos } θ} = \frac{1}{\sin θ} = \csc θ.

Step 2

4.2 sin²(180° + B) · cosec(π - B)

99%

104 rated

Answer

For the first part:

sin2(180°+B)=sin2B,\text{sin}^2(180° + B) = \text{sin}^2 B,

since (\sin(180° + B) = -\sin B). Thus:

sin2(180°+B)⋅cosec(π−B)=sin2B⋅1sin⁡(π−B)=sin2B⋅1sin⁡B=sinB.\text{sin}^2(180° + B) · \text{cosec}(π - B) = \text{sin}^2 B · \frac{1}{\sin(π - B)} = \text{sin}^2 B · \frac{1}{\sin B} = \text{sin} B.

Now, for the second part:

sec(2π−B)⋅cos(180°−B)=1cos⁡(2π−B)⋅(−cosB)=−cosBcos⁡(2π−B).\text{sec}(2π - B) · \text{cos}(180° - B) = \frac{1}{\cos(2π - B)} · (-\text{cos} B) = -\frac{\text{cos} B}{\cos(2π - B)}.

Join the NSC students using SimpleStudy...

97% of Students

Report Improved Results

98% of Students

Recommend to friends

100,000+

Students Supported

1 Million+

Questions answered

;