Prove each of the following identities:
(i) (1 – cos^2θ) cosec^2θ = 1
(ii) (1 + cot^2θ) sin^2θ = 1
(i) (sec^2θ − 1) cot^2θ = 1
(ii) (sec^2θ − 1) (cosec^2θ − 1) = 1
(iii) (1− cos^2θ) sec^2θ = tan^2θ
Prove:
\text { (i) } \sin ^{2} \theta+\frac{1}{1+\tan ^{2} \theta}=1
\text { (ii) } \frac{1}{1+\tan ^{2} \theta}+\frac{1}{1+\cot ^{2} \theta}=1
(i) (1 + cos θ) (1 − cos θ) (1 + cot^2θ) = 1
(ii) cosec θ (1 + cos θ) (cosec θ − cot θ) = 1
\text { (i) } \cot ^{2} \theta-\frac{1}{\sin ^{2} \theta}=-1
\text { (ii) } \tan ^{2} \theta-\frac{1}{\cos ^{2} \theta}=-1
\text { (iii) } \cos ^{2} \theta+\frac{1}{1+\cot ^{2} \theta}=1
\frac{1}{1+\sin \theta}+\frac{1}{1-\sin \theta}=2 \sec ^{2} \theta
(i) sec θ (1 − sin θ) (sec θ + tan θ) = 1
(ii) sin θ(1 + tan θ) + cos θ(1 + cot θ) = (sec θ + cosec θ)
\text { (i) } 1+\frac{\cot ^{2} \theta}{1+\cos e c \theta}=\operatorname{cosec} \theta
\text { (ii) } 1+\frac{\tan ^{2} \theta}{1+\sec \theta}=\sec \theta
\frac{\left(1+\tan ^{2} \theta\right) \cot \theta}{\operatorname{cosec}^{2} \theta}=\tan \theta
\frac{\tan ^{2} \theta}{1+\tan ^{2} \theta}+\frac{\cot ^{2} \theta}{1+\cot ^{2} \theta}=1
If x = a sec θ + b tan θ and y = a tan θ + b sec θ, prove that (x^2 − y^2) = (a^2 − b^2).
\begin{array}{l} \text { If }\left(\frac{x}{a} \sin \theta-\frac{y}{b} \cos \theta\right)=1 \text { and }\left(\frac{x}{a} \cos \theta+\frac{y}{b} \sin \theta\right)=1 \\ \text { prove that } \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=2 \end{array}
If (sec θ + tan θ) = m and (sec θ − tan θ) = n, show that mn = 1.
If (cosec θ + cot θ) = m and (cosec θ − cot θ) = n, show that mn = 1.
If x = a cos^3 θ and y = b sin^3 θ, prove that:
\left(\frac{x}{a}\right)^{2 / 3}+\left(\frac{y}{b}\right)^{2 / 3}=1
If (tan θ + sin θ) = m and (tan θ − sin θ) = n, prove that (m^2 − n^2)^2 = 16mn.
If (cot θ + tan θ) = m and (sec θ − cos θ) = n, prove that (m^2n)^(2/3) − (mn^2)^(2/3) = 1.
If (cosec θ − sin θ) = a^3 and (sec θ − cos θ) = b^3,
prove that: a^2b^2(a^2 + b^2) = 1.
If (2 sin θ + 3 cos θ) = 2, show that (3 sin θ − 2 cos θ) = ± 3.
Write the value of (1-sin^2 θ)sec^2 θ.
Write the value of (1-cos^2θ)cosec^2θ.
Write the value of (1+tan^2θ)cos^2θ.
Write the value of (1+cot^2θ)sin^2θ.
Write the value of sin^2θ + 1/(1+tan^2θ)
Write the value of (cot^2θ – 1/sin^2θ)
Write the value of cosec^2(90°-θ) – tan^2θ.
Write the value of sec^2θ(1+sinθ)(1-sinθ).
Write the value of cosec^2θ(1+cosθ)(1-cosθ).
Write the value of sin^2θ cos^2θ(1+tan^2θ)(1+cot^2θ).
Write the value of (1+tan^2θ)(1+sinθ)(1-sinθ).
Write the value of 3cot^2θ – 3cosec^2θ.
Write the value of 4tan^2θ – 4/cos^2θ
Write the value of (tan^2θ – sec^2θ) / (cot^2θ – cosec^2θ)
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