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patriot [66]
3 years ago
6

Verify that [tan(theta) + cot(theta)]^2 = sec^2(theta) + csc^2(theta)

Mathematics
1 answer:
Wittaler [7]3 years ago
7 0

(tanθ + cotθ)² = sec²θ + csc²θ

<u>Expand left side</u>: tan²θ + 2tanθcotθ + cot²θ

<u>Evaluate middle term</u>: 2tanθcotθ = 2*\frac{sin\theta}{cos\theta}*\frac{cos\theta}{sin\theta} = 2

⇒ tan²θ + 2+ cot²θ

= tan²θ + 1 + 1 + cot²θ

<u>Apply trig identity:</u>  tan²θ + 1 = sec²θ

⇒ sec²θ + 1 + cot²θ

<u>Apply trig identity:</u>  1 + cot²θ = csc²θ

⇒ sec²θ + csc²θ

Left side equals Right side so equation is verified


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