Find :
(a) (sin θ + cosec θ)2
(b) (cos θ + sec θ)2
Using the above results prove the following trigonometry identity :
(sin θ + cosec θ)2 + (cos θ + sec θ)2 = 7 + tan2 θ + cot2 θ
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(a) Solving,
⇒ (sin θ + cosec θ)2
⇒ sin2 θ + cosec2 θ + 2 × sin θ × cosec θ
⇒ sin2 θ + 1 + cot2 θ + 2 × sin θ × $\frac{1}{\text{sin θ}}$
⇒ sin2 θ + 1 + cot2 θ + 2
⇒ sin2 θ + cot2 θ + 3.
(b) Solving,
⇒ (cos θ + sec θ)2
⇒ cos2 θ + sec2 θ + 2 × cos θ × sec θ
⇒ cos2 θ + 1 + tan2 θ + 2 × cos θ × $\frac{1}{\text{cos θ}}$
⇒ cos2 θ + tan2 θ + 1 + 2
⇒ cos2 θ + tan2 θ + 3.
Solving,
⇒ (sin θ + cosec θ)2 + (cos θ + sec θ)2
⇒ sin2 θ + cot2 θ + 3 + cos2 θ + tan2 θ + 3.
⇒ sin2 θ + cos2 θ + cot2 θ + tan2 θ + 3 + 3
⇒ 1 + cot2 θ + tan2 θ + 3 + 3
⇒ 7 + tan2 θ + cot2 θ.
Hence, proved that (sin θ + cosec θ)2 + (cos θ + sec θ)2 = 7 + tan2 θ + cot2 θ.