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If ∆ABC is right angled at C, find the value of cos(A + B). NCERT Exemplar

If sin θ – cos θ = 0, (0 ≤ θ≤ 90°), find the value of θ.

In an acute ∆ABC, if tan (A + B – C) = 1 and sec (B + C – A) = 2, find the angles A, B and C.

If sin $(50°-\frac{3}{2}\theta )$ = cos (3θ – 50°), find the value of θ and hence evaluate
tanθ secθ sinθ – cotθ sinθ cosθ.

If A, B, C are the angles of a ∆ABC, prove that
$\tan \left(\frac{B+C}{2}\right)=\cot \frac{A}{2}$

If  sin $A=\frac{12}{13}$, what is the value of cos A?

If tan $\text{θ}=\frac{a}{b}$, find the value of sec θ.

If sin $C=\frac{15}{17}$, find the value of sin A.

In the following figure, ∆ABC is right angled at B and tan $A=\frac{4}{3}$, find the length of AC.

If 17 cos A = 8, find 15 cosec A – 8 sec B.

If $\sin \alpha =\frac{1}{2}$ and α is an acute angle, find the value of (3cosα – 4cos³α).

If 3 tan A = 4, prove that
(i) $\sqrt{\frac{\sec A-\text{cosec\hspace{0.17em}A}}{\sec A+\text{cosec\hspace{0.17em}A}}}=\frac{1}{\sqrt{7}}$.
(ii) $\sqrt{\frac{1-\sin \text{\hspace{0.17em}A}}{1+\text{cos\hspace{0.17em}A}}}=\frac{1}{2\sqrt{2}}$.

If $\sqrt{3}\sin \text{θ}=\cos \text{θ}$, find the value of $\frac{\sin \text{θ\hspace{0.17em}}\tan \text{θ}(1+\cot \text{θ})}{\sin \text{θ}+\cos \text{θ}}$.

Prove that $\frac{{\sec }^2\text{θ}-{\sin }^2\text{θ}}{{\tan }^2\text{θ}}={\text{cosec}}^2\text{θ}-{\cos }^2\text{θ}$.

Prove that (1 + tan A + cos A) (sin A – cos A) = sin A tan A – cot A cos A. CBSE 2015

If $\sec \text{θ}=\frac{4}{\sqrt{7}}$, prove that $\sqrt{\frac{2{\tan }^2\text{θ}-{\text{cosec}}^2\text{θ}}{2{\cos }^2\text{θ}-{\cot }^2\text{θ}}}=\frac{20}{7}$.

Find the value of (cosec² θ – 1) tan² θ.

Prove that $\sqrt{\frac{1+\cos A}{1-\cos A}}=\text{cosec\hspace{0.17em}A\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}cot\hspace{0.17em}\hspace{0.17em}A.}$

Show that
${(1-\sin A+\cos A)}^2=2(1-\sin A)(1+\cos A)$

Prove that $\frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta }{\sin \theta }=2\text{cosec}\theta$

Find the values of the constant a and b for which sin x x cos x (5 tan x + 2 cot x) = a + b sin² x. CBSE 2009

Prove that $(+{\tan }^2\theta )+(1+\frac{1}{{\tan }^2\theta })=\frac{1}{{\sin }^2\theta -{\sin }^4\theta }$

Prove that
(i) $(1+\cot \theta -\text{cosec}\theta )(1+\tan \theta +\sec \theta )=2.$ CBSE 2010, 08
(ii) ${\tan }^2\theta +{\cot }^2\theta +2={\sec }^2\theta +{\text{cosec}}^2\theta .$ CBSE 2010

Prove that $\frac{{\sec }^2\theta -{\sin }^2\theta }{{\tan }^2\theta }=1+{\cot }^2\theta -{\cos }^2\theta .$

Prove that $\frac{1-\cos A+\sin A}{\sin A+\cos A-1}=\frac{1+\sin A}{\cos A}.$  CBSE 2015

Prove that (cosecθ– sinθ) (secθ – cosθ) = sinθ · cosθ = $\frac{1}{\tan \theta +\cot \theta }.$ CBSE 2016

Prove that $\sqrt{\frac{\sin A+1}{1-\sin A}}+\sqrt{\frac{1-\sin A}{\sin A+1}}=2\sec A.$ CBSE 2016

Prove that ${\sec }^2\theta -\frac{{\sin }^2\theta -2{\sin }^4\theta }{2{\cos }^4\theta -{\cos }^2\theta }=1.$ CBSE 2015

Prove that $(\tan \theta +\sec \theta -1)(\tan \theta +1+\sec \theta )=\frac{2\sin \theta }{1-\sin \theta }$.  CBSE 2015

Given that 2sin A cos A +(cos A + sin A)² – (2 cos A + sin A)² = p sin² A + q. Find the value of p and q.

Prove that $\frac{\text{cosec}\theta +\cot \theta }{\text{cosec}\theta -\cot \theta }={\left(\text{cosec}\theta \text{+cot}\theta \right)}^2$ $=1+2{\cot }^2\theta +2\text{cosec}\theta \cdot \text{cot}\theta \text{.}$ CBSE 2008

Prove that q (p² – 1) = 2p, where $\sin \theta +\cos \theta =p$ and $\sec \theta +\text{cosec}\theta =q.$ NCERT Exemplar; CBSE 2016

If $\sec \theta +\tan \theta =p$, show that $\sec \theta -\tan \theta =\frac{1}{p}$. Hence find the value of $\cos \theta$ and $\sin \theta$. CBSE 2015

Prove that $\frac{\sin A}{1+\cos A}+\frac{\sin A}{1-\cos A}=\sqrt{\frac{1+\cos A}{1-\cos A}}+\sqrt{\frac{1-\cos A}{1+\cos A}}=2\text{\hspace{0.17em}cosec\hspace{0.17em}}A.$  CBSE 2015

If $\cos \theta -\sin \theta =\sqrt{2}\sin \theta$, prove that $\cos \theta +\text{sin\hspace{0.17em}}\theta =\sqrt{2}\cos \theta .$ CBSE 2004, 11

Show that $2({\sin }^6\theta +{\cos }^6\theta )-3({\sin }^4\theta +{\cos }^4\theta )+1=0.$

Given that $\sin \alpha =\frac{1}{2}$ and $\cos \beta =\frac{1}{2}$, what is the value of $(\alpha +\beta )$? NCERT Exemplar

If $4x=\text{cosec}\theta$ and $\frac{4}{x}=\cot \theta$, find the value of $4[x^2-\frac{1}{x^2}].$ CBSE 2010

Find and acute angle θ, when $\frac{\cos \theta -\sin \theta }{\cos \theta +\sin \theta }=\frac{1-\sqrt{3}}{1+\sqrt{3}}$.

If $\tan \theta =\cot ({30}^{\circ }+\theta )$, find the value of θ.

Prove that $(\tan \theta +2)(2\tan \theta +1)=5\tan \theta +2{\sec }^2\theta$. NCERT Exemplar

Prove that $\frac{\sin \theta -2{\sin }^3\theta }{2{\cos }^3\theta -\cos \theta }=\tan \theta$.

Prove that $\sec A(1-\sin A)(\sec A+\tan A)=1$.

If $\cot \theta =\frac{15}{8}$, find the value of $\frac{(2+2\sin \theta )(1-\sin \theta )}{(1+\cos \theta )(2-2\cos \theta )}$.

Find the value of $[\frac{{\sin }^2{22}^{\circ }+{\sin }^2{68}^{\circ }}{{\cos }^2{22}^{\circ }+{\cos }^2{68}^{\circ }}+{\sin }^2{63}^{\circ }+\cos {63}^{\circ }\sin {27}^{\circ }]$.  NCERT Exemplar

If $\cos A+{\cos }^{2}A=1$, find the value of ${\sin }^{2}A+{\sin }^{4}A.$

If $\sin A+{\sin }^{2}A=1$, find the value of ${\cos }^{2}A+{\cos }^{4}A.$

If $\tan A=\sqrt{2}-1$, prove that $\frac{\tan A}{1+{\tan }^{2}A}=\frac{\sqrt{2}}{4}.$

In right angled $\Delta ACB,\angle C={90}^{\circ }$, AB = 29 units and BC = 21 units. If $\angle ABC=\theta$, find ${\cos }^2\theta -{\sin }^2\theta$ and ${\sin }^2\theta +{\cos }^2\theta$.
Draw a $\Delta ACB$, right angled at C and $\angle ABC=\theta .$

In the given figure, $\Delta ABC$ is right angled at B. $\Delta BSC$ is right angled at S and $\Delta BRS$ is right angled at R, AB = 18 cm, BC = 7.5 cm, RS = 5 cm, RB = 6 cm, $\angle BSR=x^{\circ }$ and $\angle SAB=y^{\circ }$, Find

(i) $\tan x^{\circ }$
(ii) $\hspace{0.17em}\sin y^{\circ }$
(iii) $\cos y^{\circ }$

If $2{\sin }^2\theta -{\cos }^2\theta =2$, find the value of θ. NCERT Exemplar

If $A+B={90}^{\circ }$, prove that
$\sqrt{\frac{\tan A\tan B+\tan A\cot B}{\sin A\sec B}-\frac{{\sin }^{2}B}{{\cos }^{2}A}}=\tan A.$

Prove that $\frac{\sin \theta }{1-\cos \theta }=\text{cosec}\theta \text{\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}cot}\theta \text{.}$

If $\sec \theta =x+\frac{1}{4x}$, find the value of $\sec \theta +\tan \theta$.

Prove that $\sqrt{{\sec }^2\theta +{\text{cosec}}^2\theta }=\tan \theta +\cot \theta .$ NCERT Exemplar

Eliminate θ from the following equations.
(i) $x=a\sec \theta ,y=b\tan \theta$
(ii) $x=k+a\cos \theta ,y=h+b\sin \theta$

If $\sin {\theta }_1+\sin {\theta }_2+\sin {\theta }_3=3,$ $0^{\circ }<{\theta }_1,{\theta }_2,{\theta }_3\le {90}^{\circ }$, find the value of $\cos {\theta }_1+\cos {\theta }_2+\cos {\theta }_3.$

Evaluate $\frac{{\sec }^2({90}^{\circ }-\theta )-{\cot }^2\theta }{2({\sin }^2{25}^{\circ }+{\sin }^2{65}^{\circ })}+\frac{2{\cos }^2{60}^{\circ }{\tan }^2{28}^{\circ }{\tan }^2{62}^{\circ }}{\sin {30}^{\circ }\cos {60}^{\circ }}.$ CBSE 2015

Prove
$\cos \theta \sin \theta -\frac{\sin \theta \cos ({90}^{\circ }-\theta )\cos \theta }{\sec ({90}^{\circ }-\theta )}-\frac{\cos \theta \sin ({90}^{\circ }-\theta )\sin \theta }{\text{cosec}({90}^{\circ }-\theta )}\text{\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}cosec}({90}^{\circ }-\theta )=\frac{1}{\cos \theta }.$

Prove that $\frac{\text{cosec}\theta \text{+cot}\theta }{\text{cosec}\theta \text{-cot}\theta }=1+2{\cot }^2\theta +2{\text{cosec}}^2\theta \cos \theta .$

If $\tan \theta +\frac{1}{\tan \theta }=2,$ find the value of ${\cot }^2\theta +\frac{1}{{\cot }^2\theta }.$

Evaluate $\sin A\cos A-\frac{\sin A\cos ({90}^{\circ }-A)\cos A}{\sec ({90}^{\circ }-A)}-\frac{\cos A\sin ({90}^{\circ }-A)\sin A}{\text{cosec}({90}^{\circ }-A)}.$ CBSE 2015

Prove that ${(1+\tan A\tan B)}^2+{(\tan A-\tan B)}^2={\sec }^{2}A{\sec }^{2}B.$

If $a\cos \theta -b\sin \theta =c,$ prove that $a\sin \theta +b\cos \theta =\pm \sqrt{a^2+b^2-c^2}$.

Prove that ${\cot }^{2}A\left[\frac{\sec A-1}{1+\sin A}\right]+{\sec }^{2}A\left[\frac{\sin A-1}{1+\sec A}\right]=0.$

Prove that $[\frac{1}{{\sec }^2\theta -{\cos }^2\theta }+\frac{1}{{\text{cosec}}^2\theta -{\sin }^2\theta }]{\sin }^2\theta {\cos }^2\theta =\frac{1-{\sin }^2\theta {\cos }^2\theta }{2+{\sin }^2\theta {\cos }^2\theta }.$

Find the value of $\sec \theta \text{\hspace{0.17em}\hspace{0.17em}cosec}({90}^{\circ }-\theta )-\tan \theta \cot ({90}^{\circ }-\theta )\frac{+{\sin }^2{55}^{\circ }+{\sin }^2{35}^{\circ }}{\tan {10}^{\circ }\tan {20}^{\circ }\tan {60}^{\circ }\tan {70}^{\circ }\tan {80}^{\circ }}.$

Prove that $\frac{(1+\cot \theta +\tan \theta )(\sin \theta -\cos \theta )}{{\sec }^3\theta -{\text{cosec}}^3\theta }={\sin }^2\theta {\cos }^2\theta .$ CBSE 2008

If tan A = a tan B and sin A = b sin B, prove that ${\cos }^{2}A=\frac{b^2-1}{a^2-1}.$

If $\text{cosec\hspace{0.17em}}\theta -\sin \theta =m$ and $\sec \theta -\cos \theta =n,$ then prove that ${\left(m^{2}n\right)}^{2/3}+{\left(m{n}^2\right)}^{2/3}=1.$

Soni was trying hard to prove $a\cos \theta -b\sin \theta =\sqrt{a^2+b^2-c^2}$, when $a\sin \theta +b\cos \theta =c.$ Her classmate Swati gave her a hint of squaring both sides of $a\sin \theta +b\cos \theta =c$ and proceed further.
Following her hint, Soni was able to solve the problem and she thanks Swati for this hint.
(i) Write the solution of the problem.
(ii) What value is depicted here?