Question

Prove that (sin θ + cosec θ)² + (cos θ + sec θ)² - (tan² θ + cot² θ) = 7

Answer 1

Solution is in the attachment

Answer 2

Answer:

Step-by-step explanation:

Prove that (sin θ + cosec θ)² + (cos θ + sec θ)² - (tan² θ + cot² θ) = 7

As we know that $cosec\:\theta=\frac{1}{sin\:\theta} \\\\sec\:\theta=\frac{1}{cos\:\theta}\\\\cot\:\theta=\frac{1}{tan\:\theta}\\\\so\\\\(sin\:\theta+\frac{1}{sin\:\theta})^{2}+(cos\:\theta+\frac{1}{cos\:\theta})^{2} -(tan^{2}\:\theta+\frac{1}{cot^{2}\:\theta})\\\\\\sin^{2}\theta+\frac{1}{sin^{2}\theta} +2\:sin\:\theta\frac{1}{sin\:\theta}+cos^{2}\theta+\frac{1}{cos^{2}\theta} +2\:cos\:\theta\frac{1}{cos\:\theta}-tan^{2}\:\theta-cot^{2}\:\theta$

$=5+\frac{1}{sin^{2}\theta}+\frac{1}{cos^{2}\theta}-tan^{2}\theta-cot^{2}\theta\\\\=5+cosec^{2}\theta+sec^{2}\theta-tan^{2}\theta-cot^{2}\theta\\\\=5+1+cot^{2}\theta+1+tan^{2}\theta-tan^{2}\theta-cot^{2}\theta\\\\=5+1+1\\\\=7$