Prove the following trigonometric identities:
(sec2 θ − 1)(cosec2 θ − 1) = 1
tan θ + 1/ tan θ = sec θ cosec θ
(1 + tan2 θ)(1 – sin θ)(1 + sin θ) = 1
(1 – cos2 A) cosec2 A = 1
(1 + cot2 A) sin2 A = 1
tan2 θ cos2 θ = 1 − cos2 θ
Prove
\frac{\left(1+\cot ^{2} \theta\right) \tan \theta}{\sec ^{2} \theta}=\cot \theta
(cosec θ + sin θ)(cosec θ – sin θ) = cot2θ + cos2θ
cosec θ √(1 – cos2 θ) = 1
cos θ/ (1 – sin θ) = (1 + sin θ)/ cos θ
tan2 θ − sin2 θ = tan2 θ sin2 θ
cos θ/ (1 + sin θ) = (1 – sin θ)/ cos θ
cos2 θ + 1/(1 + cot2 θ) = 1
sin2 A + 1/(1 + tan 2 A) = 1
\sqrt{\frac{1-\cos \theta}{1+\cos \theta}}=\operatorname{cosec} \theta-\cot \theta
1 – cos θ/ sin θ = sin θ/ 1 + cos θ
(1 – sin θ) / (1 + sin θ) = (sec θ – tan θ)2
(sec θ + cos θ) (sec θ – cos θ) = tan2 θ + sin2 θ
sec A(1- sin A) (sec A + tan A) = 1
sin2 A cot2 A + cos2 A tan2 A = 1
\frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}=2 \sec \theta
\frac{1+\tan ^{2} \theta}{1+\cot ^{2} \theta}=\tan ^{2} \theta
\frac{1+\sec \theta}{\sec \theta}=\frac{\sin ^{2} \theta}{1-\cos \theta}
sec6 θ = tan6 θ + 3 tan2 θ sec2 θ + 1
cosec6 θ = cot6 θ + 3cot2 θ cosec2 θ + 1
\frac{1+\cos A}{\sin ^{2} A}=\frac{1}{1-\cos A}
1+\frac{\cos A}{\sin A}=\frac{\sin A}{1-\cos A}
\frac{\sec \theta-1}{\sec \theta+1}=\left(\frac{\sin \theta}{1+\cos \theta}\right)^{2}
\frac{1-\cos A}{1+\cos A}=(\cot A-\operatorname{cosec} A)^{2}
\frac{1}{\sec A-1}+\frac{1}{\sec A+1}=2 \operatorname{cosec} A \cot A
\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}=\sin A+\cos A
(\sec A-\tan A)^{2}=\frac{1-\sin A}{1+\sin A}
\frac{(1+\sin \theta)^{2}+(1-\sin \theta)^{2}}{2 \cos ^{2} \theta}=\frac{1+\sin ^{2} \theta}{1-\sin ^{2} \theta}
sin θ/ (1 – cos θ) = cosec θ + cot θ
If cos θ = 4/5, find all other trigonometric ratios of angle θ.
If sin θ = 1/√2, find all other trigonometric ratios of angle θ.
\text { If } \tan \theta=\frac{1}{\sqrt{2}} \text {, find the value of } \frac{\operatorname{cosec}^{2} \theta-\sec ^{2} \theta}{\operatorname{cosec}^{2} \theta+\cot ^{2} \theta}
\text { If } \operatorname{cosec} A=\sqrt{2} \text {, find the value of } \frac{2 \sin ^{2} A+3 \cot ^{2} A}{4\left(\tan ^{2} A-\cos ^{2} A\right)}
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